# Idea

The von Neumann hierarchy is a way of “building up” all pure sets recursively, starting with the empty set, and indexed by the ordinal numbers.

# Definition

Using transfinite recursion?, define a hierarchy of well-founded sets ${V}_{\alpha }$, where $\alpha \in \mathrm{Ord}$ is an ordinal number, as follows:

• ${V}_{0}=\varnothing$
• ${V}_{\alpha +1}=P\left({V}_{\alpha }\right)$ (the power set of ${V}_{\alpha }$)
• ${V}_{\alpha }={\cup }_{\beta <\alpha }{V}_{\beta }$ if $\alpha$ is a limit ordinal.

The formula for $0$ is actually a special case of the formula for a limit ordinal. Alternatively, you can do them all at once:

• ${V}_{\alpha }={\cup }_{\beta <\alpha }P\left({V}_{\beta }\right)$

The axiom of foundation in ZFC is equivalent to the statement that every set is an element of ${V}_{\alpha }$ for some ordinal $\alpha$. The rank of a set $x$ is defined to be the least $\alpha$ for which $x\in {V}_{\alpha }$ (this is well-defined since the ordinals are well-ordered).

Revised on April 11, 2009 05:34:03 by Toby Bartels (71.104.234.95)