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:

$\displaystyle V \left({x}\right) = \begin{cases} \varnothing & : x = 0 \\ & \\ \mathcal P \left({ V \left({n}\right) }\right) & : x = n^+ \\ & \\ \displaystyle \bigcup_{y \mathop \in x} V \left({y}\right) & : x \in \operatorname{Lim} \\ \end{cases}$

where:

$\mathcal P \left({x}\right)$ 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.