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			{{For|the symbol|Infinity symbol}}
			{{For|other uses of "Infinity" and "Infinite"}}
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	For the symbol, see Infinity symbol.
			{{short description|Mathematical concept}}
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
			]]]
			'''Infinity''' represents something that is boundless or endless, or else something that is larger than any ] or ].<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#infinite|title=The Definitive Glossary of Higher Mathematical Jargon — Infinite|date=2019-08-01|website=Math Vault|language=en-US|access-date=2019-11-15}}</ref> It is often denoted by the ] '''{{math|∞}}'''.
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.		Since the time of the ], the ] was the subject of many discussions among philosophers. In the 17th century, with the introduction of the ]<ref name=":1">{{Cite web|url=https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf|title=The History of Infinity|last=Allen|first=Donald|date=2003|website=Texas A&M Mathematics|access-date=2019-11-15}}</ref> and the ], mathematicians began to work with ] and what some mathematicians (including ] and ])<ref name="Jesseph" /> regarded as infinitely small quantities, but infinity continued to be associated with endless processes.<ref>The ] status of infinitesimals was unclear, but only some mathematicians regarded infinitesimal as a quantity that is smaller (in magnitude) than any positive number. Others viewed it either as an artefact that makes computation easier or as a small quantity that can be made smaller and smaller until the quantity in which it is involved reaches eventually a ].{{citation needed|date=November 2019}}</ref> As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.<ref name=":1" /> At the end of the 19th century, ] enlarged the mathematical study of infinity by studying ]s and ]s, showing that they can be of various sizes.<ref name=":1" /><ref>{{cite book
			|title=The Princeton Companion to Mathematics
			|first1=Timothy
			|last1=Gowers
			|first2=June
			|last2=Barrow-Green
			|first3=Imre
			|last3=Leader
			|publisher=Princeton University Press
			|year=2008
			|isbn=978-0-691-11880-2
			|page=616
			|url=https://books.google.com/books?id=LmEZMyinoecC
			|url-status=live
			|archive-url=https://web.archive.org/web/20160603163923/https://books.google.com/books?id=LmEZMyinoecC
			|archive-date=2016-06-03
			}} {{webarchive|url=https://web.archive.org/web/20160501221357/https://books.google.com/books?id=LmEZMyinoecC&pg=PA616 |date=2016-05-01 }}
			</ref> For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the ] of the line) is larger than the number of ]s.<ref>{{harvnb|Maddox|2002|loc=pp. 113–117}}</ref> In this usage, infinity is a mathematical concept, and infinite ]s can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.		The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of ], on which most of modern mathematics can be developed, is the ], which guarantees the existence of infinite sets.<ref name=":1" /> The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as ] that may seem to have nothing to do with them. For example, ] of ] implicitly relies on the existence of ]<ref>{{cite journal |title=What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory |last=McLarty |first=Colin |journal=The Bulletin of Symbolic Logic |volume=16 |issue=3 |year=2010 |pages=359–377 |doi=10.2178/bsl/1286284558}}</ref> for solving a long-standing problem that is stated in terms of ].
	In physics and cosmology, whether the Universe is infinite is an open question.		In ] and ], ] is an open question.
			==History==
			{{Further|Infinity (philosophy)}}
			Ancient cultures had various ideas about the nature of infinity. The ] and ] did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
			===Early Greek===
	Contents
			The earliest recorded idea of infinity may be that of ] (c. 610 – c. 546 BC) a ] Greek philosopher. He used the word ], which means "unbounded", "indefinite", and perhaps can be translated as "infinite".<ref name=":1" /><ref>{{harvnb|Wallace|2004|p=44}}</ref>
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
			Aristotle (350 BC) distinguished ''potential infinity'' from '']'', which he regarded as impossible due to the various paradoxes it seemed to produce.<ref>{{cite book |author=Aristotle |url=http://classics.mit.edu/Aristotle/physics.3.iii.html |translator-last1=Hardie|translator-first1=R. P. |translator-last2=Gaye|translator-first2=R. K. |at=Book 3, Chapters 5–8|title=Physics|publisher=The Internet Classics Archive}}</ref> It has been argued that, in line with this view, the ] Greeks had a "horror of the infinite"<ref>{{cite journal|author=Nicolas D. Goodman|journal=Constructive Mathematics. Lecture Notes in Mathematics|title=Reflections on Bishop's philosophy of mathematics|publisher=Springer|editor1-last=Richman|editor1-first=F.|volume=873|year=1981}}</ref><ref>Maor, p. 3</ref> which would, for example, explain why ] (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."<ref>{{cite book|last1=Heath|first1=Sir Thomas Little|last2=Heiberg|first2=Johan Ludvig|author-link1=Thomas Heath (classicist)|title=The Thirteen Books of Euclid's Elements|volume=v. 2|publisher=The University Press|year=1908|page=412 (Book IX, Proposition 20)|url=https://books.google.com/books?id=lxkPAAAAIAAJ&q=BOOK+IX&pg=RA2-PA112}}.</ref> It has also been maintained, that, in proving ], Euclid "was the first to overcome the horror of the infinite".<ref>{{cite book|last=Hutten|first=Earnest H.|title=The Origins of Science: An Inquiry into the Foundations of Western Thought|url=https://archive.org/details/originsofscience0000hutt_n9u7|url-access=registration|year=1962|publisher=George Allen & Unwin Ltd.|page=}}</ref> There is a similar controversy concerning Euclid's ], sometimes translated
	Early Greek
			:If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.<ref>{{cite book|author=Euclid |orig-year=c. 300 BC|translator-last1=Fitzpatrick |translator-first1=Richard |title=Euclid's Elements of Geometry |url=http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf|year=2008 |isbn=978-0-6151-7984-1 |page=6 (Book I, Postulate 5)}}</ref>
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
			Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",<ref>{{cite book|last1=Heath|first1=Sir Thomas Little|last2=Heiberg|first2=Johan Ludvig|author-link1=Thomas Heath (classicist)|title=The Thirteen Books of Euclid's Elements|volume=v. 1|publisher=The University Press|year=1908|url=https://books.google.com/books?id=dkk6AQAAMAAJ&q=right+angles+infinite&pg=PR8|page=212}}</ref> thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.<ref>{{cite book|last=Drozdek|first=Adam|title=''In the Beginning Was the'' Apeiron'': Infinity in Greek Philosophy''|year=2008|isbn=978-3-515-09258-6|publisher=Franz Steiner Verlag|location=Stuttgart, Germany}}
			</ref>
			===Zeno: Achilles and the tortoise===
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
			{{Main|Zeno's paradoxes#Achilles and the tortoise}}
			] (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,<ref name="Zeno's paradoxes">{{cite web|url=https://plato.stanford.edu/entries/paradox-zeno/ |title=Zeno's Paradoxes |date=October 15, 2010 |website=Stanford University |access-date=April 3, 2017}}</ref> especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by ] as "immeasurably subtle and profound".<ref>{{harvnb|Russell|1996|p=347}}</ref>
			] races a tortoise, giving the latter a head start.
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
			:Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
			:Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Zeno: Achilles and the tortoise		:Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
			:Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.		Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.		Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.		Zeno was not attempting to make a point about infinity. As a member of the ] school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,		Finally, in 1821, ] provided both a satisfactory definition of a limit and a proof that, for 0 < ''x'' < 1,
			:''a'' + ''ax'' + ''ax''<sup>2</sup> + ''ax''<sup>3</sup> + ''ax''<sup>4</sup> + ''ax''<sup>5</sup> + '''· · ·''' = {{resize|110%|{{sfrac|''a''|1&minus;''x''}}}} .<ref>{{cite book|last=Cauchy|first=Augustin-Louis|author-link=Augustin-Louis Cauchy|access-date=October 12, 2019|title=Cours d'Analyse de l'École Royale Polytechnique|year=1821|publisher=Libraires du Roi & de la Bibliothèque du Roi|url=https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1|page=124}}</ref>
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
			Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with ''a''&nbsp;=&nbsp;10 seconds and ''x''&nbsp;=&nbsp;0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
			:10 + 0.1 + 0.001 + 0.00001 + '''· · ·''' = {{sfrac|10|1&minus;0.01}} = {{sfrac|10|0.99}} = 10 {{sfrac|10|99}} seconds.
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
			===Early Indian===
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
			The ] text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: ], innumerable, and infinite. Each of these was further subdivided into three orders:<ref>{{cite book|author=Ian Stewart|title=Infinity: a Very Short Introduction|url=https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|year=2017|publisher=Oxford University Press|isbn=978-0-19-875523-4|page=117|url-status=live|archive-url=https://web.archive.org/web/20170403200429/https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|archive-date=April 3, 2017}}</ref>
			* Enumerable: lowest, intermediate, and highest
			* Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
			* Infinite: nearly infinite, truly infinite, infinitely infinite
			===17th century===
	Mathematics
			In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, ] first used the notation <math>\infty</math> for such a number in his ''De sectionibus conicis,''<ref>{{Cite book|url=https://books.google.com/books?id=OQZxHpG2y3UC&q=infinity|title=A History of Mathematical Notations|last=Cajori|first=Florian|publisher=Cosimo, Inc.|year=2007|isbn=9781602066854|volume=1|pages=214|language=en}}</ref> and exploited it in area calculations by dividing the region into ] strips of width on the order of <math>\tfrac{1}{\infty}.</math><ref>{{harvnb|Cajori|1993|loc=Sec. 421, Vol. II, p. 44}}</ref> But in ''Arithmetica infinitorum'' (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."<ref>{{harvnb|Cajori|1993|loc=Sec. 435, Vol. II, p. 58}}</ref>
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
			In 1699, ] wrote about equations with an infinite number of terms in his work '']''.<ref>{{cite book |title=Landmark Writings in Western Mathematics 1640-1940 |first1=Ivor |last1=Grattan-Guinness |publisher=Elsevier |year=2005 |isbn=978-0-08-045744-4 |page=62 |url=https://books.google.com/books?id=UdGBy8iLpocC |url-status=live |archive-url=https://web.archive.org/web/20160603085825/https://books.google.com/books?id=UdGBy8iLpocC |archive-date=2016-06-03 }} </ref>
	Mathematics is the science of the infinite.
			==Mathematics==
	Symbol
			] opened a mathematico-philosophic address given in 1930 with:<ref>{{citation|first=Hermann|last=Weyl|title=Levels of Infinity / Selected Writings on Mathematics and Philosophy|editor=Peter Pesic|year=2012|publisher=Dover|isbn=978-0-486-48903-2|page=17}}</ref>
	Main article: Infinity symbol
			{{quote|text=Mathematics is the science of the infinite.}}
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
			===Symbol===
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
			{{Main|Infinity symbol}}
			The infinity symbol <math>\infty</math> (sometimes called the ]) is a mathematical symbol representing the concept of infinity. The symbol is encoded in ] at {{unichar|221E|infinity|html=}}<ref>{{Cite web|url=https://www.compart.com/en/unicode/U+221E|title=Unicode Character "∞" (U+221E)|last=AG|first=Compart|website=Compart.com|language=en|access-date=2019-11-15}}</ref> and in ] as <code>\infty</code>.<ref>{{Cite web|url=https://oeis.org/List_of_LaTeX_mathematical_symbols|title=List of LaTeX mathematical symbols - OeisWiki|website=oeis.org|access-date=2019-11-15}}</ref>
			It was introduced in 1655 by ],<ref>{{citation
	Calculus
			| last = Scott
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
			| first = Joseph Frederick
			| edition = 2
			| isbn = 978-0-8284-0314-6
			| page = 24
			| publisher = ]
			| title = The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)
			| url = https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24
			| year = 1981
			| url-status=live
			| archive-url = https://web.archive.org/web/20160509151853/https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24
			| archive-date = 2016-05-09
			}}</ref><ref>{{citation
			| last = Martin-Löf | first = Per | author-link = Per Martin-Löf
			| contribution = Mathematics of infinity
			| doi = 10.1007/3-540-52335-9_54
			| location = Berlin
			| mr = 1064143
			| pages = 146–197
			| publisher = Springer
			| series = Lecture Notes in Computer Science
			| title = COLOG-88 (Tallinn, 1988)
			| volume = 417
			| year = 1990| isbn = 978-3-540-52335-2 }}</ref> and since its introduction, it has also been used outside mathematics in modern mysticism<ref>{{citation|title=Dreams, Illusion, and Other Realities|first=Wendy Doniger|last=O'Flaherty|publisher=University of Chicago Press|year=1986|isbn=978-0-226-61855-5|page=243|url=https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|url-status=live|archive-url=https://web.archive.org/web/20160629143323/https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|archive-date=2016-06-29}}</ref> and literary ].<ref>{{citation|title=Nabokov: The Mystery of Literary Structures|first=Leona|last=Toker|publisher=Cornell University Press|year=1989|isbn=978-0-8014-2211-9|page=159|url=https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159|url-status=live|archive-url=https://web.archive.org/web/20160509095701/https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159|archive-date=2016-05-09}}</ref>
			===Calculus===
	Real analysis
			], one of the co-inventors of ], speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the ].<ref>{{cite SEP |url-id=continuity |title=Continuity and Infinitesimals |last=Bell |first=John Lane |author-link=John Lane Bell}}</ref><ref name="Jesseph">{{cite journal
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
			|last=Jesseph
			|first=Douglas Michael
			|year=1998
			|title=Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes
			|journal=]
			|volume=6
			|issue=1&2
			|pages=6–40
			|issn=1063-6145
			|oclc=42413222
			|url=http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html
			|access-date=1 November 2019
			|archive-date=15 February 2010
			|archive-url=https://www.webcitation.org/5nZWht6FE?url=http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html
			|url-status=dead
			}}</ref>
			====Real analysis====
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
			In ], the symbol <math>\infty</math>, called "infinity", is used to denote an unbounded ].<ref>{{harvnb|Taylor|1955|loc=p. 63}}</ref> The notation <math>x \rightarrow \infty</math> means that&nbsp;''<math>x</math>'' increases without bound, and <math>x \to -\infty</math> means that&nbsp;''<math>x</math>'' decreases without bound. For example, if <math>f(t)\ge 0</math> for every&nbsp;''<math>t</math>'', then<ref>These uses of infinity for integrals and series can be found in any standard calculus text, such as, {{harvnb|Swokowski|1983|pp=468–510}}</ref>
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
			* <math>\int_{a}^{b} f(t)\, dt = \infty</math> means that <math>f(t)</math> does not bound a finite area from <math>a</math> to <math>b.</math>
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
			* <math>\int_{-\infty}^{\infty} f(t)\, dt = \infty</math> means that the area under <math>f(t)</math> is infinite.
	Infinity can also be used to describe infinite series, as follows:
			* <math>\int_{-\infty}^{\infty} f(t)\, dt = a</math> means that the total area under <math>f(t)</math> is finite, and is equal to <math>a.</math>
			Infinity can also be used to describe ], as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.		* <math>\sum_{i=0}^{\infty} f(i) = a</math> means that the sum of the infinite series ] to some real value <math>a.
			</math>
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
			* <math>\sum_{i=0}^{\infty} f(i) = \infty</math> means that the sum of the infinite series properly ] to infinity, in the sense that the partial sums increase without bound.<ref>{{Cite web|url=http://mathonline.wikidot.com/properly-divergent-sequences|title=Properly Divergent Sequences - Mathonline|website=mathonline.wikidot.com|access-date=2019-11-15}}</ref>
			In addition to defining a limit, infinity can be also used as a value in the extended real number system.<ref name=":0" /> Points labeled <math>+\infty</math> and <math>-\infty</math> can be added to the ] of the real numbers, producing the two-point ] of the real numbers. Adding algebraic properties to this gives us the ]s.<ref>{{citation
	Complex analysis
			| last1 = Aliprantis
			| first1 = Charalambos D.
			| last2 = Burkinshaw
			| first2 = Owen
			| edition = 3rd
			| isbn = 978-0-12-050257-8
			| location = San Diego, CA
			| mr = 1669668
			| page = 29
			| publisher = Academic Press, Inc.
			| title = Principles of Real Analysis
			| url = https://books.google.com/books?id=m40ivUwAonUC&pg=PA29
			| year = 1998
			| url-status=live
			| archive-url = https://web.archive.org/web/20150515120230/https://books.google.com/books?id=m40ivUwAonUC&pg=PA29
			| archive-date = 2015-05-15
			}}</ref> We can also treat <math>+\infty</math> and <math>-\infty</math> as the same, leading to the ] of the real numbers, which is the ].<ref>{{harvnb|Gemignani|1990|loc=p. 177}}</ref> ] also refers to a ] in plane geometry, a ] in three-dimensional space, and a ] for general ], each consisting of ].<ref>{{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|title=Projective Geometry / from foundations to applications|year=1998|publisher=Cambridge University Press|isbn=978-0-521-48364-3|page=27}}</ref>
			====Complex analysis====
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
			], the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the ].]]
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
			In ] the symbol <math>\infty</math>, called "infinity", denotes an unsigned infinite ]. <math>x \rightarrow \infty</math> means that the magnitude&nbsp;<math>|x|</math> of&nbsp;''<math>x</math>'' grows beyond any assigned value. A ] can be added to the complex plane as a ] giving the ] of the complex plane.<ref>{{Cite web|url=http://mathworld.wolfram.com/ExtendedComplexPlane.html|title=Extended Complex Plane|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-15}}</ref> When this is done, the resulting space is a one-dimensional ], or ], called the extended complex plane or the ]. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables ], namely <math>z/0 = \infty</math> for any nonzero complex number&nbsp;''<math>z</math>''. In this context, it is often useful to consider ]s as maps into the Riemann sphere taking the value of <math>\infty</math> at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of ]s (see ]).
	Nonstandard analysis
			===Nonstandard analysis===
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
			]
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
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	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
	Mathematics
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
	Mathematics is the science of the infinite.
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
	Calculus
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
	Complex analysis
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
	Nonstandard analysis
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
	Set theory
	Main articles: Cardinality and Ordinal number
	One-to-one correspondence between an infinite set and its proper subset
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
	Cardinality of the continuum
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
	Geometry
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
	In particular, in modern mathematics, lines are infinite sets.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
	Cosmology
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
	Logic
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	Jump to navigationJump to search
	For the symbol, see Infinity symbol.
	For other uses of "Infinity" and "Infinite", see Infinity (disambiguation).
	The infinity symbol
	Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.
	Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
	The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
	In physics and cosmology, whether the Universe is infinite is an open question.
	Contents
	1 History
	1.1 Early Greek
	1.2 Zeno: Achilles and the tortoise
	1.3 Early Indian
	1.4 17th century
	2 Mathematics
	2.1 Symbol
	2.2 Calculus
	2.2.1 Real analysis
	2.2.2 Complex analysis
	2.3 Nonstandard analysis
	2.4 Set theory
	2.4.1 Cardinality of the continuum
	2.5 Geometry
	2.6 Infinite dimension
	2.7 Fractals
	2.8 Mathematics without infinity
	3 Physics
	3.1 Cosmology
	4 Logic
	5 Computing
	6 Arts, games, and cognitive sciences
	7 See also
	8 References
	8.1 Bibliography
	8.2 Sources
	9 External links
	History
	Further information: Infinity (philosophy)
	Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
	Early Greek
	The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
	Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated
	If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.
	Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
	Zeno: Achilles and the tortoise
	Main article: Zeno's paradoxes § Achilles and the tortoise
	Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
	Achilles races a tortoise, giving the latter a head start.
	Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
	Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
	Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
	Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
	Etc.
	Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.
			The original formulation of ] by ] and Gottfried Leibniz used ] quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various ]s, including ] and ]. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a ]; there is no equivalence between them as with the Cantorian ]. For example, if&nbsp;H is an infinite number in this sense, then H&nbsp;+&nbsp;H&nbsp;=&nbsp;2H and H&nbsp;+&nbsp;1 are distinct infinite numbers. This approach to ] is fully developed in {{harvtxt|Keisler|1986}}.
	Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.
			===Set theory===
	Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
			{{Main|Cardinality|Ordinal number}}
			]
	a + ax + ax2 + ax3 + ax4 + ax5 + · · · =
	a
	/
	1−x
	.
	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
			A different form of "infinity" are the ] and ] infinities of set theory—a system of ]s first developed by ]. In this system, the first transfinite cardinal is ] (<span style="font-family:'Cambria Math';"><big>ℵ</big><sub>0</sub></span>), the cardinality of the set of ]s. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, ], ] and others—using the idea of collections or sets.<ref name=":1" />
	10 + 0.1 + 0.001 + 0.00001 + · · · =
	10
	/
	1−0.01
	=
	10
	/
	0.99
	= 10
	10
	/
	99
	seconds.
	Early Indian
	The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
			Dedekind's approach was essentially to adopt the idea of ] as a standard for comparing the size of sets, and to reject the view of Galileo (derived from ]) that the whole cannot be the same size as the part (however, see ] where he concludes that positive ] are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its ] parts; this notion of infinity is called ]. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".{{citation needed|date=April 2017}}
	Enumerable: lowest, intermediate, and highest
	Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
	Infinite: nearly infinite, truly infinite, infinitely infinite
	17th century
	In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation {\displaystyle \infty }\infty for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of {\displaystyle {\tfrac {1}{\infty }}.}{\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."
			Cantor defined two kinds of infinite numbers: ]s and ]s. Ordinal numbers characterize ] sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite ]s which are maps from the positive ] leads to ] from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is ]. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called '']''. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.<ref>{{Cite web|url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html|title=Infinity|website=math.dartmouth.edu|access-date=2019-11-16}}</ref><ref>{{cite book
	In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
			|title=The Infinite
			|first1=A.W.
			|last1=Moore
			|publisher=Routledge
			|year=1991}}
			</ref>{{page needed|date=June 2014}} Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.{{citation needed|date=April 2017}}
			====Cardinality of the continuum====
	Mathematics
			{{Main|Cardinality of the continuum}}
	Hermann Weyl opened a mathematico-philosophic address given in 1930 with:
			One of Cantor's most important results was that the cardinality of the continuum <math>\mathbf c</math> is greater than that of the natural numbers <math>{\aleph_0}</math>; that is, there are more real numbers '''R''' than natural numbers '''N'''. Namely, Cantor showed that <math>\mathbf{c} = 2^{\aleph_0} > {\aleph_0}</math> (see ] or ]).<ref>{{Cite journal| last = Dauben
	Mathematics is the science of the infinite.
			| first = Joseph
			| title = Georg Cantor and the Battle for Transfinite Set Theory
			| url = http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf
			| journal = 9th ACMS Conference Proceedings
			| year = 1993
			| page = 4
			}}</ref>
			The ] states that there is no ] between the cardinality of the reals and the cardinality of the natural numbers, that is, <math>\mathbf{c} = \aleph_1 = \beth_1 </math> (see ]). This hypothesis cannot be proved or disproved within the widely accepted ], even assuming the ].<ref>{{harvnb|Cohen|1963|p=1143}}</ref>
	Symbol
	Main article: Infinity symbol
	The infinity symbol {\displaystyle \infty }\infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML &#8734; · &infin;) and in LaTeX as \infty.
			] can be used to show not only that the number of points in a ] is equal to the number of points in any ] of that line, but also that this is equal to the number of points on a plane and, indeed, in any ] space.{{citation needed|date=April 2017}}
	It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
			], showing that there are as many points in a one-dimensional line as in a two-dimensional square.]]
	Calculus
			The first of these results is apparent by considering, for instance, the ] function, which provides a ] between the ] (−π/2, π/2) and&nbsp;'''R''' (see also ]). The second result was proved by Cantor in&nbsp;1878, but only became intuitively apparent in&nbsp;1890, when ] introduced the ]s, curved lines that twist and turn enough to fill the whole of any square, or ], or ], or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.<ref>{{harvnb|Sagan|1994|pp=10–12}}</ref>
	Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
			===Geometry===
	Real analysis
	In real analysis, the symbol {\displaystyle \infty }\infty , called "infinity", is used to denote an unbounded limit. The notation {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that {\displaystyle x}x increases without bound, and {\displaystyle x\to -\infty }x\to -\infty means that {\displaystyle x}x decreases without bound. For example, if {\displaystyle f(t)\geq 0}{\displaystyle f(t)\geq 0} for every {\displaystyle t}t, then
			Until the end of the 19th century, infinity was rarely discussed in ], except in the context of processes that could be continued without any limit. For example, a ] was what is now called a ], with the proviso that one can extend it as far as one wants; but extending it ''infinitely'' was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the ] of ''a point'' that satisfies some property" (singular), where modern mathematicians would generally say "the set of ''the points'' that have the property" (plural).
	{\displaystyle \int _{a}^{b}f(t)\,dt=\infty }{\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that {\displaystyle f(t)}f(t) does not bound a finite area from {\displaystyle a}a to {\displaystyle b.}{\displaystyle b.}
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty } means that the area under {\displaystyle f(t)}f(t) is infinite.
	{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a} means that the total area under {\displaystyle f(t)}f(t) is finite, and is equal to {\displaystyle a.}a.
	Infinity can also be used to describe infinite series, as follows:
			One of the rare exceptions of a mathematical concept involving ] was ], where ] are added to the ] for modeling the ] effect that shows ] intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a ], two distinct ] intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
	{\displaystyle \sum _{i=0}^{\infty }f(i)=a}{\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value {\displaystyle a.}{\displaystyle a.}
	{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty } means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
	In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat {\displaystyle +\infty }+\infty and {\displaystyle -\infty }-\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
			Before the use of ] for the ], points and lines were viewed as distinct entities, and a point could be ''located on a line''. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as ''the set of its points'', and one says that a point ''belongs to a line'' instead of ''is located on a line'' (however, the latter phrase is still used).
	Complex analysis
			In particular, in modern mathematics, lines are ''infinite sets''.
	By stereographic projection, the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the Riemann sphere.
	In complex analysis the symbol {\displaystyle \infty }\infty , called "infinity", denotes an unsigned infinite limit. {\displaystyle x\rightarrow \infty }x\rightarrow \infty means that the magnitude {\displaystyle |x|}|x| of {\displaystyle x}x grows beyond any assigned value. A point labeled {\displaystyle \infty }\infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely {\displaystyle z/0=\infty }z/0=\infty for any nonzero complex number {\displaystyle z}z. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of {\displaystyle \infty }\infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).
			===Infinite dimension===
	Nonstandard analysis
			The ]s that occur in classical ] have always a finite ], generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in ] where ]s are generally vector spaces of infinite dimension.
			In topology, some constructions can generate ]s of infinite dimension. In particular, this is the case of ]s.
	Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
	The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).
			===Fractals===
	Set theory
			The structure of a ] object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such ] with an infinite perimeter and finite area is the ].{{citation needed|date=April 2017}}
	Main articles: Cardinality and Ordinal number
			===Mathematics without infinity===
	One-to-one correspondence between an infinite set and its proper subset
			] was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the ] called ], an extreme form of mathematical philosophy in the general philosophical and mathematical schools of ] and ].<ref>{{harvnb|Kline|1972|pp=1197–1198}}</ref>
	A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.
			==Physics==
	Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
			In ], approximations of ]s are used for ] measurements and ]s are used for ] measurements (i.e., counting). Concepts of infinite things such as an infinite ] exist, but there are no experimental means to generate them.<ref> {{webarchive|url=https://web.archive.org/web/20130124011604/http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf |date=2013-01-24 }} – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.</ref>
	Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
			===Cosmology===
	Cardinality of the continuum
			The first published proposal that the universe is infinite came from Thomas Digges in 1576.<ref>John Gribbin (2009), ''In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality'', {{isbn|978-0-470-61352-8}}. p.&nbsp;88</ref> Eight years later, in 1584, the Italian philosopher and astronomer ] proposed an unbounded universe in ''On the Infinite Universe and Worlds'': "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."<ref>{{cite book |title=Alien Life Imagined: Communicating the Science and Culture of Astrobiology |journal=Physics Today |volume=67 |issue=6 |edition=illustrated |first1=Mark |last1=Brake |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-49129-7 |page=63 |url=https://books.google.com/books?id=sWGqzfL0snEC|bibcode=2014PhT....67f..49S |doi=10.1063/PT.3.2420 }} </ref>
	Main article: Cardinality of the continuum
	One of Cantor's most important results was that the cardinality of the continuum {\displaystyle \mathbf {c} }\mathbf {c} is greater than that of the natural numbers {\displaystyle {\aleph _{0}}}{\aleph _{0}}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}\mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}} (see Cantor's diagonal argument or Cantor's first uncountability proof).
			] have long sought to discover whether infinity exists in our physical ]: Are there an infinite number of stars? Does the universe have infinite volume? Does space ]? This is an open question of ]. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar ]. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.<ref>{{cite book |title=In Quest of the Universe |edition=illustrated |first1=Theo |last1=Koupelis |first2=Karl F. |last2=Kuhn |publisher=Jones & Bartlett Learning |year=2007 |isbn=978-0-7637-4387-1 |page=553 |url=https://books.google.com/books?id=6rTttN4ZdyoC}} </ref>
	The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}\mathbf {c} =\aleph _{1}=\beth _{1} (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.
			The curvature of the universe can be measured through ] in the spectrum of the ]. To date, analysis of the radiation patterns recorded by the ] spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.<ref name="NASA_Shape">{{cite web| title=Will the Universe expand forever?| url=http://map.gsfc.nasa.gov/universe/uni_shape.html| publisher=NASA| date=24 January 2014| access-date=16 March 2015| url-status=live| archive-url=https://web.archive.org/web/20120601032707/http://map.gsfc.nasa.gov/universe/uni_shape.html| archive-date=1 June 2012}}</ref><ref name="Fermi_Flat">{{cite web| title=Our universe is Flat| url=http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725| publisher=FermiLab/SLAC| date=7 April 2015| url-status=live| archive-url=https://web.archive.org/web/20150410200411/http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725| archive-date=10 April 2015}}</ref><ref>{{cite journal|title=Unexpected connections|author=Marcus Y. Yoo|journal=Engineering & Science|volume=LXXIV1|date=2011|page=30}}</ref>
	Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.
			However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is ] and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.<ref>{{cite book|last=Weeks|first=Jeffrey|title=The Shape of Space|year=2001|publisher=CRC Press|isbn=978-0-8247-0709-5|url-access=registration|url=https://archive.org/details/shapeofspace0000week}}</ref>
			The concept of infinity also extends to the ] hypothesis, which, when explained by astrophysicists such as ], posits that there are an infinite number and variety of universes.<ref>Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.</ref>
	The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line as in a two-dimensional square.
	The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.
			==Logic==
	Geometry
			In ], an ] argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form&nbsp;A) no such series exists or (form&nbsp;B) were it to exist, the thesis would lack the role (e.g.,&nbsp;of justification) that it is supposed to play."<ref>''Cambridge Dictionary of Philosophy'', Second Edition, p.&nbsp;429</ref>
	Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).
			==Computing==
	One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.
			The ] standard (IEEE 754) specifies a positive and a negative infinity value (and also ] values). These are defined as the result of ], ], and other exceptional operations.{{citation needed|date=April 2017}}
			Some ]s, such as ]<ref>{{cite book|last=Gosling|first=James |display-authors=etal |title=The Java Language Specification|publisher=Oracle America, Inc.|location=California|date=27 July 2012|edition=Java SE 7|chapter=4.2.3.|access-date=6 September 2012|chapter-url=http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3|url-status=live|archive-url=https://web.archive.org/web/20120609071157/http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3|archive-date=9 June 2012}}</ref> and ],<ref>
	Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).
			{{cite book
			|last= Stokes
			|first= Roger
			|title= Learning J
			|date= July 2012
			|chapter= 19.2.1
			|chapter-url= http://www.rogerstokes.free-online.co.uk/19.htm#10
			|access-date= 6 September 2012
			|url-status=dead
			|archive-url= https://web.archive.org/web/20120325064205/http://www.rogerstokes.free-online.co.uk/19.htm#10
			|archive-date= 25 March 2012
			}}</ref> allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as ], as they compare (respectively) greater than or less than all other values. They have uses as ]s in ]s involving ], ], or ].{{citation needed|date=April 2017}}
			In languages that do not have greatest and least elements, but do allow ] of ]s, it is possible for a programmer to ''create'' the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point ], the infinity values may still be accessible and usable as the result of certain operations.{{citation needed|date=April 2017}}
	In particular, in modern mathematics, lines are infinite sets.
			In programming, an ] is a ] whose exit condition is never satisfied, thus executing indefinitely.
	Infinite dimension
	The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.
			==Arts, games, and cognitive sciences==
	In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.
			] artwork utilizes the concept of ]s, roughly corresponding to mathematical ], located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.<ref>{{cite book
			|title=Mathematics for the nonmathematician
			|first1=Morris
			|last1=Kline
			|publisher=Courier Dover Publications
			|year=1985
			|isbn=978-0-486-24823-3
			|page=
			|url=https://archive.org/details/mathematicsforno00klin
			|url-access=registration
			}}, {{webarchive|url=https://web.archive.org/web/20160516173217/https://books.google.com/books?id=f-e0bro-0FUC&pg=PA229 |date=2016-05-16 }}
			</ref> Artist ] is specifically known for employing the concept of infinity in his work in this and other ways.{{citation needed|date=April 2017}}
			Variations of ] played on an unbounded board are called ].<ref> {{webarchive|url=https://web.archive.org/web/20170402082426/http://www.chessvariants.com/boardrules.dir/infinite.html |date=2017-04-02 }} An infinite chess scheme.</ref><ref> {{webarchive|url=https://web.archive.org/web/20170407211614/https://www.youtube.com/watch?v=PN-I6u-AxMg |date=2017-04-07 }} PBS Infinite Series,with academic sources by J. Hamkins (infinite chess: {{cite arxiv |eprint=1302.4377 |last1=Evans |first1=C.D.A |title=Transfinite game values in infinite chess |author2=Joel David Hamkins |class=math.LO |year=2013 }} and {{cite arxiv |eprint=1510.08155 |last1=Evans |first1=C.D.A |title=A position in infinite chess with game value $ω^4$ |author2=Joel David Hamkins |author3=Norman Lewis Perlmutter |class=math.LO |year=2015 }}).</ref>
	Fractals
	The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.
			] ] considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity&nbsp;(BMI), defined as the ever-increasing sequence <1,2,3,...>.{{citation needed|date=April 2017}}
	Mathematics without infinity
	Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.
			The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.{{Citation needed|date = January 2015}}
	Physics
	In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.
			==See also==
	Cosmology
			{{Div col|colwidth=20em}}
	The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			{{Div col end}}
			==References==
	Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
			{{reflist}}
			===Bibliography===
	The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
			{{Refbegin}}
			* {{citation|first=Florian|last=Cajori|title=A History of Mathematical Notations (Two Volumes Bound as One)|year=1993|orig-year=1928 & 1929|publisher=Dover|isbn=978-0-486-67766-8|url=https://archive.org/details/historyofmathema00cajo_0}}
			* {{citation|first=Michael C.|last=Gemignani|title=Elementary Topology|edition=2nd|publisher=Dover|year=1990|isbn=978-0-486-66522-1}}
			* {{citation|first=H. Jerome|last=Keisler|author-link=Howard Jerome Keisler|title=Elementary Calculus: An Approach Using Infinitesimals|edition=2nd|year=1986|url=http://www.math.wisc.edu/~keisler/calc.html}}
			<!--
			* H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
			-->
			* {{citation|first=Randall B.|last=Maddox|title=Mathematical Thinking and Writing: A Transition to Abstract Mathematics|publisher=Academic Press|year= 2002|isbn=978-0-12-464976-7}}
			* {{citation |title=Mathematical Thought from Ancient to Modern Times |last=Kline |first=Morris |author-link=Morris Kline |year=1972 |publisher=Oxford University Press |location= New York|isbn=978-0-19-506135-2 |pages=1197–1198}}
			* {{citation |title=The Principles of Mathematics |last=Russell |first=Bertrand |author-link=Bertrand Russell |year=1996 |orig-year=1903 |publisher=Norton |location=New York|isbn=978-0-393-31404-5 |oclc=247299160}}
			* {{citation | first=Hans | last=Sagan | title=Space-Filling Curves | publisher=Springer | year=1994 | isbn=978-1-4612-0871-6}}
			* {{citation|first=Earl W.|last=Swokowski|title=Calculus with Analytic Geometry|edition=Alternate|year=1983|publisher=Prindle, Weber & Schmidt|isbn=978-0-87150-341-1|url=https://archive.org/details/calculuswithanal00swok}}
			* {{citation|first=Angus E.|last=Taylor|title=Advanced Calculus|year=1955|publisher=Blaisdell Publishing Company}}
			* {{citation | first=David Foster | last=Wallace | author-link=David Foster Wallace | title=Everything and More: A Compact History of Infinity | publisher=Norton, W.W. & Company, Inc. | year=2004 | isbn=978-0-393-32629-1}}
			{{Refend}}
			===Sources===
	However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.
			{{Refbegin}}
			*{{cite book | first=Amir D. |last=Aczel | title=The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity | publisher=Pocket Books|place=New York | year=2001 | isbn=978-0-7434-2299-4}}
			*] (2000). '''', .
			* Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
			*{{Citation |last=Cohen |first=Paul |title=The Independence of the Continuum Hypothesis |doi=10.1073/pnas.50.6.1143 |journal=] |volume= 50|issue=6 |pages=1143–1148 |year=1963 |pmid=16578557 |pmc=221287|bibcode=1963PNAS...50.1143C }}.
			*{{cite book | first=L.C. |last=Jain | title=Exact Sciences from Jaina Sources | year=1982}}
			* Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", ''Indian Journal of History of Science''.
			*{{cite book | first=George G. |last=Joseph | title=The Crest of the Peacock: Non-European Roots of Mathematics | edition=2nd | publisher=] | year=2000 | isbn= 978-0-14-027778-4}}
			* H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
			*{{cite book | author=Eli Maor | title=To Infinity and Beyond | publisher=Princeton University Press | year=1991 | isbn=978-0-691-02511-7| author-link=Eli Maor |url=https://books.google.com/books?id=lXjF7JnHQoIC&q=To+Infinity+and+beyond}}
			* O'Connor, John J. and Edmund F. Robertson (1998). , '']''.
			* O'Connor, John J. and Edmund F. Robertson (2000). , ''MacTutor History of Mathematics archive''.
			* Pearce, Ian. (2002). , ''MacTutor History of Mathematics archive''.
			*{{cite book | author=Rucker, Rudy | title=Infinity and the Mind: The Science and Philosophy of the Infinite | publisher=Princeton University Press | year=1995 | isbn=978-0-691-00172-2| author-link=Rudy Rucker }}
			*{{cite journal | first=Navjyoti |last=Singh | title=Jaina Theory of Actual Infinity and Transfinite Numbers | journal=Journal of the Asiatic Society | volume=30 | year=1988}}<!-- {{cite book | author=Navjyoti Singh | title=Jaina Theory of Actual Infinity and Transfinite Numbers | journal=Vaishali Institute Research Bulletin| volume=5 | year=1986}}-->
			{{Refend}}
			==External links==
	The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.
			{{Wiktionary}}
			{{Wikibooks|Infinity is not a number}}
			{{commons category}}
			{{wikiquote}}
			* {{cite IEP |url-id=infinite |title=The Infinite}}
			* {{In Our Time|Infinity|p0054927|Infinity}}
			* '''', by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to ''Infinite Reflections'', below. A concise introduction to Cantor's mathematics of infinite sets.
			* '''', by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
			* {{cite web|last=Grime|first=James|title=Infinity is bigger than you think|url=http://www.numberphile.com/videos/countable_infinity.html|work=Numberphile|publisher=]|access-date=2013-04-06|archive-url=https://web.archive.org/web/20171022173525/http://www.numberphile.com/videos/countable_infinity.html|archive-date=2017-10-22|url-status=dead}}
			*
			* John J. O'Connor and Edmund F. Robertson (1998). , '']''.
			* John J. O'Connor and Edmund F. Robertson (2000). , ''MacTutor History of Mathematics archive''.
			* Ian Pearce (2002). , ''MacTutor History of Mathematics archive''.
			*
			*
			* (compilation of articles about infinity in physics, mathematics, and philosophy)
			{{Infinity}}
	Logic
			{{Large numbers}}
	In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
			{{Analysis-footer}}
			{{Authority control}}
	Computing
	The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
	Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least
	]		]
	]		]

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Revision as of 20:37, 19 February 2021

For the symbol, see Infinity symbol. For other uses of "Infinity" and "Infinite", see Infinity (disambiguation). Mathematical concept

Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol ∞.

Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.

In physics and cosmology, whether the Universe is infinite is an open question.

History

Further information: Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".

Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving this theorem, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated

If a straight line falling across two straight lines makes internal angles on the same side less than two right angles, then the two straight lines, being produced to infinity, meet on that side that the is less than two right angles.

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.

Zeno: Achilles and the tortoise

Main article: Zeno's paradoxes § Achilles and the tortoise

Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".

Achilles races a tortoise, giving the latter a head start.

Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.

Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.

Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.

Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.

Etc.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,

a + ax + ax + ax + ax + ax + · · · = ⁠a/1−x⁠ .

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him

10 + 0.1 + 0.001 + 0.00001 + · · · = ⁠10/1−0.01⁠ = ⁠10/0.99⁠ = 10 ⁠10/99⁠ seconds.

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

• Enumerable: lowest, intermediate, and highest
• Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
• Infinite: nearly infinite, truly infinite, infinitely infinite

17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation ${\displaystyle \infty }$ for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of ${\displaystyle {\tfrac {1}{\infty }}.}$ But in Arithmetica infinitorum (also in 1655), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c."

In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.

Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:

Mathematics is the science of the infinite.

Symbol

Main article: Infinity symbol

The infinity symbol ${\displaystyle \infty }$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (&infin;) and in LaTeX as \infty.

It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.

Calculus

Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.

Real analysis

In real analysis, the symbol ${\displaystyle \infty }$, called "infinity", is used to denote an unbounded limit. The notation ${\displaystyle x\rightarrow \infty }$ means that ${\displaystyle x}$ increases without bound, and ${\displaystyle x\to -\infty }$ means that ${\displaystyle x}$ decreases without bound. For example, if ${\displaystyle f(t)\geq 0}$ for every ${\displaystyle t}$, then

• ${\displaystyle \int _{a}^{b}f(t)\,dt=\infty }$ means that ${\displaystyle f(t)}$ does not bound a finite area from ${\displaystyle a}$ to ${\displaystyle b.}$
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty }$ means that the area under ${\displaystyle f(t)}$ is infinite.
• ${\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}$ means that the total area under ${\displaystyle f(t)}$ is finite, and is equal to ${\displaystyle a.}$

Infinity can also be used to describe infinite series, as follows:

• ${\displaystyle \sum _{i=0}^{\infty }f(i)=a}$ means that the sum of the infinite series converges to some real value ${\displaystyle a.}$
• ${\displaystyle \sum _{i=0}^{\infty }f(i)=\infty }$ means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.

In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.

Complex analysis

In complex analysis the symbol ${\displaystyle \infty }$, called "infinity", denotes an unsigned infinite limit. ${\displaystyle x\rightarrow \infty }$ means that the magnitude ${\displaystyle |x|}$ of ${\displaystyle x}$ grows beyond any assigned value. A point labeled ${\displaystyle \infty }$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely ${\displaystyle z/0=\infty }$ for any nonzero complex number ${\displaystyle z}$. In this context, it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of ${\displaystyle \infty }$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview).

Nonstandard analysis

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).

Set theory

Main articles: Cardinality and Ordinal number

A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ₀), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets.

Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum ${\displaystyle \mathbf {c} }$ is greater than that of the natural numbers ${\displaystyle {\aleph _{0}}}$; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}}$ (see Cantor's diagonal argument or Cantor's first uncountability proof).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, ${\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}$ (see Beth one). This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory, even assuming the Axiom of Choice.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.

Geometry

Until the end of the 19th century, infinity was rarely discussed in geometry, except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment, with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural).

One of the rare exceptions of a mathematical concept involving actual infinity was projective geometry, where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry.

Before the use of set theory for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be located on a line. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points, and one says that a point belongs to a line instead of is located on a line (however, the latter phrase is still used).

In particular, in modern mathematics, lines are infinite sets.

Infinite dimension

The vector spaces that occur in classical geometry have always a finite dimension, generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension.

In topology, some constructions can generate topological spaces of infinite dimension. In particular, this is the case of iterated loop spaces.

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters, and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake.

Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.

Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.

Logic

In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."

Computing

The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.

Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

Perspective artwork utilizes the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

Variations of chess played on an unbounded board are called infinite chess.

Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.

The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.

See also

• 0.999...
• Aleph number
• Ananta
• Exponentiation
• Indeterminate form
• Infinite monkey theorem
• Infinite set
• Infinitesimal
• Paradoxes of infinity
• Supertask
• Surreal number

References

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Bibliography

• Cajori, Florian (1993) , A History of Mathematical Notations (Two Volumes Bound as One), Dover, ISBN 978-0-486-67766-8
• Gemignani, Michael C. (1990), Elementary Topology (2nd ed.), Dover, ISBN 978-0-486-66522-1
• Keisler, H. Jerome (1986), Elementary Calculus: An Approach Using Infinitesimals (2nd ed.)
• Maddox, Randall B. (2002), Mathematical Thinking and Writing: A Transition to Abstract Mathematics, Academic Press, ISBN 978-0-12-464976-7
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press, pp. 1197–1198, ISBN 978-0-19-506135-2
• Russell, Bertrand (1996) , The Principles of Mathematics, New York: Norton, ISBN 978-0-393-31404-5, OCLC 247299160
• Sagan, Hans (1994), Space-Filling Curves, Springer, ISBN 978-1-4612-0871-6
• Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1
• Taylor, Angus E. (1955), Advanced Calculus, Blaisdell Publishing Company
• Wallace, David Foster (2004), Everything and More: A Compact History of Infinity, Norton, W.W. & Company, Inc., ISBN 978-0-393-32629-1

Sources

• Aczel, Amir D. (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 978-0-7434-2299-4.
• D.P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
• Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
• Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.
• Jain, L.C. (1982). Exact Sciences from Jaina Sources.
• Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
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• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 978-0-691-02511-7.
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• Pearce, Ian. (2002). 'Jainism', MacTutor History of Mathematics archive.
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• Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers". Journal of the Asiatic Society. 30.

External links

• "The Infinite". Internet Encyclopedia of Philosophy.
• Infinity on In Our Time at the BBC
• A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
• Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
• Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran. Archived from the original on 2017-10-22. Retrieved 2013-04-06.
• Hotel Infinity
• John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive.
• John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive.
• Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
• Source page on medieval and modern writing on Infinity
• The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
• Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)

v t e Infinity (∞)
History	Ananta (infinite) Apeiron Controversy over Cantor's theory Galileo's paradox Hilbert's paradox of the Grand Hotel Infinity (philosophy) Paradoxes of infinity Paradoxes of set theory
Branches of mathematics	Complex analysis Internal set theory Nonstandard analysis Set theory Synthetic differential geometry
Formalizations of infinity	0.999... Absolute infinite Actual infinity Aleph number Beth number Cardinal numbers Cardinality of the continuum Dedekind-infinite set Directed infinity Division by zero (Complex infinity) Epsilon number Gimel function Hilbert space Hyperreal numbers Infinite set Infinitesimal Ordinal numbers Point at infinity Regular cardinal Sphere at infinity (Kleinian group) Supertask Surreal numbers Transfinite numbers
Geometries	Differential geometry of surfaces Möbius plane Möbius transformation Riemannian manifold
Mathematicians	Georg Cantor David Hilbert Gottfried Wilhelm Leibniz August Ferdinand Möbius Bernhard Riemann Abraham Robinson

v t e Large numbers
Examples in numerical order	Hundred Thousand Ten thousand Hundred thousand Million Billion Trillion Quadrillion Quintillion Sextillion Septillion Octillion Nonillion Decillion Eddington number Googol Shannon number Googolplex Skewes's number Moser's number Graham's number TREE(3) SSCG(3) BH(3) Rayo's number Infinity
Expression methods	Notations Scientific notation Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation Operators Hyperoperation Tetration Pentation Ackermann function Grzegorczyk hierarchy Fast-growing hierarchy
Related articles (alphabetical order)	Busy beaver Extended real number line Indefinite and fictitious numbers Infinitesimal Largest known prime number List of numbers Long and short scales Number systems Number names Orders of magnitude Power of two Power of three Power of 10 Sagan Unit
Names History

v t e Major topics in mathematical analysis
Calculus: Integration Differentiation Differential equations ordinary partial stochastic Fundamental theorem of calculus Calculus of variations Vector calculus Tensor calculus Matrix calculus Lists of integrals Table of derivatives
Real analysis Complex analysis Hypercomplex analysis (quaternionic analysis) Functional analysis Fourier analysis Least-squares spectral analysis Harmonic analysis P-adic analysis (P-adic numbers) Measure theory Representation theory Functions Continuous function Special functions Limit Series Infinity
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