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 {\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}} or that W 0 ≠ ∅ {\displaystyle W_{0}\neq \emptyset }. Wikipedia