Cumulative hierarchy

Short description: Family of sets indexed by ordinal numbers

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 $W_{\alpha +1}\subseteq {𝒫}(W_{\alpha })$ or that $W_0\ne \mathrm{\varnothing }$.

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.

The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy ${\mathrm{V}}_{\alpha }$ of the von Neumann universe with ${\mathrm{V}}_{\alpha +1}={𝒫}(W_{\alpha })$ introduced by (Zermelo 1930).

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

• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. 
• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre". Fundamenta Mathematicae 16: 29–47. doi:10.4064/fm-16-1-29-47. https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/16/0/92877/uber-grenzzahlen-und-mengenbereiche.