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 ${\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })}$.^{[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 von Neumann universe, which has ${\displaystyle W_{\alpha +1}={\mathcal {P}}(W_{\alpha })}$.

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

Examples

• 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.