# 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:

$\map V x = \begin{cases} \O & : x = 0 \\ & \\ \powerset {\map V n} & : x = n^+ \\ & \\ \displaystyle \bigcup_{y \mathop \in x} \map V y & : x \in \operatorname {Lim} \\ \end{cases}$

where:

$\powerset x$ denotes the power set of $x$
$\operatorname {Lim}$ denotes the set of limit ordinals.

## Also see

• Results about the von Neumann hierarchy can be found here.

## Source of Name

This entry was named for John von Neumann.