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 αV_\alpha, where αOrd\alpha\in\mathbf{Ord} is an ordinal number, as follows:

The formula for 00 is actually a special case of the formula for a limit ordinal. Alternatively, you can do them all at once:

The axiom of foundation in ZFC is equivalent to the statement that every set is an element of V αV_\alpha for some ordinal α\alpha. The rank of a set xx is defined to be the least α\alpha for which xV αx\in V_\alpha (this is well-defined since the ordinals are well-ordered).