In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. 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.[1] 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 hierarchy Vα , called the cumulative hierarchy, based on their rank.