Misplaced Pages

In mathematics, particularly in set theory, if ${\displaystyle \kappa }$ is a regular uncountable cardinal then ${\displaystyle \operatorname {club} (\kappa ),}$ the filter of all sets containing a club subset of ${\displaystyle \kappa ,}$ is a ${\displaystyle \kappa }$-complete filter closed under diagonal intersection called the club filter.

To see that this is a filter, note that ${\displaystyle \kappa \in \operatorname {club} (\kappa )}$ since it is thus both closed and unbounded (see club set). If ${\displaystyle x\in \operatorname {club} (\kappa )}$ then any subset of ${\displaystyle \kappa }$ containing ${\displaystyle x}$ is also in ${\displaystyle \operatorname {club} (\kappa ),}$ since ${\displaystyle x,}$ and therefore anything containing it, contains a club set.

It is a ${\displaystyle \kappa }$-complete filter because the intersection of fewer than ${\displaystyle \kappa }$ club sets is a club set. To see this, suppose ${\displaystyle \langle C_{i}\rangle _{i<\alpha }}$ is a sequence of club sets where ${\displaystyle \alpha <\kappa .}$ Obviously ${\displaystyle C=\bigcap C_{i}}$ is closed, since any sequence which appears in ${\displaystyle C}$ appears in every ${\displaystyle C_{i},}$ and therefore its limit is also in every ${\displaystyle C_{i}.}$ To show that it is unbounded, take some ${\displaystyle \beta <\kappa .}$ Let ${\displaystyle \langle \beta _{1,i}\rangle }$ be an increasing sequence with ${\displaystyle \beta _{1,1}>\beta }$ and ${\displaystyle \beta _{1,i}\in C_{i}}$ for every ${\displaystyle i<\alpha .}$ Such a sequence can be constructed, since every ${\displaystyle C_{i}}$ is unbounded. Since ${\displaystyle \alpha <\kappa }$ and ${\displaystyle \kappa }$ is regular, the limit of this sequence is less than ${\displaystyle \kappa .}$ We call it ${\displaystyle \beta _{2},}$ and define a new sequence ${\displaystyle \langle \beta _{2,i}\rangle }$ similar to the previous sequence. We can repeat this process, getting a sequence of sequences ${\displaystyle \langle \beta _{j,i}\rangle }$ where each element of a sequence is greater than every member of the previous sequences. Then for each ${\displaystyle i<\alpha ,}$ ${\displaystyle \langle \beta _{j,i}\rangle }$ is an increasing sequence contained in ${\displaystyle C_{i},}$ and all these sequences have the same limit (the limit of ${\displaystyle \langle \beta _{j,i}\rangle }$). This limit is then contained in every ${\displaystyle C_{i},}$ and therefore ${\displaystyle C,}$ and is greater than ${\displaystyle \beta .}$

To see that ${\displaystyle \operatorname {club} (\kappa )}$ is closed under diagonal intersection, let ${\displaystyle \langle C_{i}\rangle ,}$ ${\displaystyle i<\kappa }$ be a sequence of club sets, and let ${\displaystyle C=\Delta _{i<\kappa }C_{i}.}$ To show ${\displaystyle C}$ is closed, suppose ${\displaystyle S\subseteq \alpha <\kappa }$ and ${\displaystyle \bigcup S=\alpha .}$ Then for each ${\displaystyle \gamma \in S,}$ ${\displaystyle \gamma \in C_{\beta }}$ for all ${\displaystyle \beta <\gamma .}$ Since each ${\displaystyle C_{\beta }}$ is closed, ${\displaystyle \alpha \in C_{\beta }}$ for all ${\displaystyle \beta <\alpha ,}$ so ${\displaystyle \alpha \in C.}$ To show ${\displaystyle C}$ is unbounded, let ${\displaystyle \alpha <\kappa ,}$ and define a sequence ${\displaystyle \xi _{i},}$ ${\displaystyle i<\omega }$ as follows: ${\displaystyle \xi _{0}=\alpha ,}$ and ${\displaystyle \xi _{i+1}}$ is the minimal element of ${\displaystyle \bigcap _{\gamma <\xi _{i}}C_{\gamma }}$ such that ${\displaystyle \xi _{i+1}>\xi _{i}.}$ Such an element exists since by the above, the intersection of ${\displaystyle \xi _{i}}$ club sets is club. Then ${\displaystyle \xi =\bigcup _{i<\omega }\xi _{i}>\alpha }$ and ${\displaystyle \xi \in C,}$ since it is in each ${\displaystyle C_{i}}$ with ${\displaystyle i<\xi .}$

See also

• Clubsuit – in set theory, the combinatorial principle that, for every stationary 𝑆⊂ω₁, there exists a sequence of sets 𝐴_𝛿 (𝛿∈𝑆) such that 𝐴_𝛿 is a cofinal subset of 𝛿 and every unbounded subset of ω₁ is contained in some 𝐴_𝛿Pages displaying wikidata descriptions as a fallback
• Filter (mathematics) – In mathematics, a special subset of a partially ordered set
• Stationary set – Set-theoretic concept

References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.

This article incorporates material from club filter on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

v t e Mathematical logic
General	Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory
Theorems (list)  and paradoxes	Gödel's completeness and incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem, paradox and diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox
Logics	Traditional Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus
Set theory	Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps and cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary Set theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive
Formal systems (list), language and syntax	Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list Example axiomatic systems (list) of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica
Proof theory	Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories
Model theory	Interpretation function of models Model equivalence finite saturated spectrum submodel Non-standard model of arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity
Computability theory	Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory
Related	Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline Logicism Mathematical object Philosophy of mathematics Supertask
Mathematics portal

v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

• Set theory