Definition:Von Neumann Hierarchy

Definition
Let $U$ denote the universal class.

The von Neumann hierarchy is a mapping $V: \operatorname{On} \to U$ on the ordinals, defined via transfinite recursion:


 * $\displaystyle V \left({x}\right) = \begin{cases}

\varnothing & : x = 0 \\ & \\ \mathcal P \left({ V \left({n}\right) }\right) & : x = n^+ \\ & \\ \bigcup_{y \mathop \in x} V \left({y}\right) & : x \in \mathrm{Lim} \\ \end{cases}$ where:
 * $\mathcal P \left({x}\right)$ denotes the power set of $x$
 * $\mathrm{Lim}$ denotes the set of limit ordinals.