Cumulative hierarchy

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 ${\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }}$

Some authors additionally require that $W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })$ or that $W_{0}\neq \emptyset$.

The union ${\textstyle W=\bigcup _{\alpha \in \mathrm {On} }W_{\alpha }}$ 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 standard cumulative hierarchy $\mathrm {V} _{\alpha }$ of the von Neumann universe with $\mathrm {V} _{\alpha +1}={\mathcal {P}}(W_{\alpha })$ 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. Zbl 1007.03002.
• 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.