Editing Von Neumann universe (section)

===Rank of a set===

The '''rank''' of a set ''S'' is the smallest α such that <math>S \subseteq V_\alpha \,.</math> In other words, <math>\mathcal{P} (V_\alpha) </math> is the set of sets with rank ≤α. The stage ''V''<sub>α</sub> can also be characterized as the set of sets with rank strictly less than α, regardless of whether α is 0, a successor ordinal, or a limit ordinal:

<math display="block">V_\alpha := \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta).</math>

This gives an equivalent definition of ''V''<sub>α</sub> by transfinite recursion.

Substituting the above definition of ''V''<sub>α</sub> back into the definition of the rank of a set gives a self-contained recursive definition:

{{block indent| The rank of a set is the smallest ordinal number strictly greater than the rank of all of its members.}}

In other words,

<math display="block">\operatorname{rank} (S) = \bigcup \{ \operatorname{rank} (z) + 1 \mid z \in S \}.</math>