Cumulative hierarchy

Template:Short description In mathematics, specifically set theory, a cumulative hierarchy is a family of sets $W_{\alpha }$ indexed by ordinals $\alpha$ such that

• $W_{\alpha }\subseteq W_{\alpha +1}$
• If $\lambda$ is a limit ordinal, then $W_{\lambda }=\bigcup _{\alpha <\lambda }{W}_{\alpha }$

Some authors additionally require that $W_{\alpha +1}\subseteq 𝒫(W_{\alpha })$.Template:Cn

The union $W=\bigcup _{\alpha \in \mathrm{O}\mathrm{n}}{W}_{\alpha }$ of the sets of a cumulative hierarchy is often used as a model of set theory.Template:Cn

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

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 $W$ of the hierarchy also holds in some stages $W_{\alpha }$.

Examples

• The von Neumann universe is built from a cumulative hierarchy ${\mathrm{V}}_{\alpha }$.
• The sets ${\mathrm{L}}_{\alpha }$ 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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