Cumulative hierarchy

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

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

Some authors additionally require that ${\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })}$ or that ${\displaystyle W_{0}\neq \emptyset }$.^{[citation needed]}

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.^{[citation needed]}

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

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

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

• 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.