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 the Second Principle of 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.

Also see

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

Source of Name

This entry was named for John von Neumann.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.9$