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Cumulative hierarchy

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Family of sets indexed by ordinal numbers

In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W α {\displaystyle W_{\alpha }} indexed by ordinals α {\displaystyle \alpha } such that

  • W α ⊆ W α + 1 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}}
  • If λ {\displaystyle \lambda } is a limit ordinal, then W λ = ⋃ α < λ W α {\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }}

Some authors additionally require that W α + 1 ⊆ P ( W α ) {\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })} .

The union W = ⋃ α ∈ O n W α {\textstyle W=\bigcup _{\alpha \in \mathrm {On} }W_{\alpha }} of the sets of a cumulative hierarchy is often used as a model of set theory.

The phrase "the cumulative hierarchy" usually refers to the von Neumann universe, which has W α + 1 = P ( W α ) {\displaystyle W_{\alpha +1}={\mathcal {P}}(W_{\alpha })} .

Reflection principle

A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union W {\displaystyle W} of the hierarchy also holds in some stages W α {\displaystyle W_{\alpha }} .

Examples

  • The von Neumann universe is built from a cumulative hierarchy V α {\displaystyle \mathrm {V} _{\alpha }} .
  • The sets L α {\displaystyle \mathrm {L} _{\alpha }} of the constructible universe form a cumulative hierarchy.
  • The Boolean-valued models constructed by forcing are built using a cumulative hierarchy.
  • The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.

References

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