nLab
von Neumann hierarchy

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:

• $V_0=\varnothing$
• $V_{\alpha +1}=P(V_{\alpha })$ (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:

• $V_{\alpha }={\cup }_{\beta <\alpha }P(V_{\beta })$

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).