page_title: Von Neumann universe
section_title: Definition
hierarchical_section_title: Von Neumann universe / Definition
context_page_description: 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, is often used to provide an interpretation or motivation of the axioms of ZFC.
  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. 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.
context_section_description: The cumulative hierarchy is a collection of sets Vα
  indexed by the class of ordinal numbers; in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α. Vα may be defined by transfinite recursion as follows:
  Let V₀ be the empty set:
  For any ordinal number β, let Vβ+1 be the power set of Vβ:
  For any limit ordinal λ, let Vλ be the union of all the V-stages so far:
  A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".
  The sets  Vα are called stages or ranks.
  The class V is defined to be the union of all the V-stages:
  An equivalent definition sets
  for each ordinal α, where  is the powerset of .
  The rank of a set S is the smallest α such that  Another way to calculate rank is:
  .
caption: An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.