Definition:Von Neumann Hierarchy

Definition
Let $U$ denote the universal class.

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


 * $\map V x = \begin{cases}

\O & : x = 0 \\ & \\ \powerset {\map V n} & : x = n^+ \\ & \\ \ds \bigcup_{y \mathop \in x} \map V y & : x \in \operatorname {Lim} \\ \end{cases}$ where:
 * $\powerset x$ denotes the power set of $x$
 * $\operatorname {Lim}$ denotes the set of limit ordinals.