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

• W α ⊆ W α + 1 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}}
• If λ {\displaystyle \lambda } is a limit ordinal, then W λ = ⋃ α < λ W α {\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }}

Some authors additionally require that W α + 1 ⊆ P ( W α ) {\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })} .

The union W = ⋃ α ∈ O n W α {\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 von Neumann universe, which has W α + 1 = P ( W α ) {\displaystyle W_{\alpha +1}={\mathcal {P}}(W_{\alpha })} .