Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted 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 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 a transfinite hierarchy, called the cumulative hierarchy, based on their rank.

The cumulative hierarchy is a collection of sets V_α indexed by the class of ordinal numbers; in particular, V_α is the set of all sets having ranks less than α. Thus there is one set V_α for each ordinal number α. V_α may be defined by transfinite recursion as follows:

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 V_α".

The sets V_α 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 ${\displaystyle {\mathcal {P}}(X)\!}$ is the powerset of ${\displaystyle X}$.

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