W\alpha
\alpha
W\alpha\subseteqW\alpha
λ
Some authors additionally require that
W\alpha\subseteqlP(W\alpha)
W0\ne\emptyset
The union 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
V\alpha
V\alpha=lP(W\alpha)
A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union
W
W\alpha
V\alpha
L\alpha