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 ${\displaystyle W_{0}\neq \emptyset }$.

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 ${\displaystyle \mathrm {V} _{\alpha }}$ of the von Neumann universe with ${\displaystyle \mathrm {V} _{\alpha +1}={\mathcal {P}}(W_{\alpha })}$ introduced by Zermelo (1930).