Editing Von Neumann universe (section)

{{short description|Set theory concept}}
In [[set theory]] and related branches of [[mathematics]], the '''von Neumann universe''', or '''von Neumann hierarchy of sets''', denoted by '''''V''''', is the [[class (set theory)|class]] of [[hereditary set|hereditary]] [[well-founded set]]s. This collection, which is formalized by [[Zermelo–Fraenkel set theory]] (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after [[John von Neumann]], although it was first published by [[Ernst Zermelo]] in 1930.

The '''rank''' of a well-founded set is defined inductively as the smallest [[ordinal number]] greater than the ranks of all members of the set.<ref>{{harvnb|Mirimanoff|1917}}; {{harvnb|Moore|2013|pp=261–262}}; {{harvnb|Rubin|1967|p=214}}.</ref> In particular, the rank of the [[empty set]] is zero, and every ordinal has a rank equal to itself. The sets in ''V'' are divided into the [[Transfinite induction|transfinite]] hierarchy ''V<sub>α</sub>{{space|hair}}'', called '''the cumulative hierarchy''', based on their rank.