von Neumann hierarchy

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.

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

- $V_0 = \emptyset$
- $V_{\alpha+1} = P(V_\alpha)$ (the power set of $V_\alpha$)
- $V_\alpha = \cup_{\beta\lt\alpha} V_\beta$ 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:

- $V_\alpha = \cup_{\beta \lt \alpha} P(V_\beta)$

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).