Von Neumann universe

In set theory and related fields, the von Neumann universe—also called the von Neumann hierarchy of sets and written as V—is the collection of all hereditary well-founded sets. This class is defined using Zermelo–Fraenkel set theory (ZFC) and frequently serves to illustrate or justify the axioms of ZFC. Though named after John von Neumann, Ernst Zermelo first published the idea in 1930.

The rank of a well-founded set is built up inductively: it is the smallest ordinal number that is greater than the ranks of every member of the set. So the empty set has rank zero, and each ordinal has a rank equal to itself. The sets in V are arranged into a transfinite hierarchy called the cumulative hierarchy, labeled Vα, based on their rank.

The cumulative hierarchy consists of sets Vα indexed by all ordinal numbers. Each Vα contains every set whose rank is less than α. The definition proceeds by transfinite recursion: V0 is the empty set; for any ordinal β, Vβ+1 is the power set of Vβ; and for a limit ordinal λ, Vλ is the union of all earlier stages. A key point is that there exists a single formula in the language of ZFC that says “α is an ordinal and x belongs to Vα.” The Vα are called stages or ranks, and V itself is the union of all these stages.

A set S has rank equal to the smallest α such that S is a subset of Vα. Equivalently, the power set of Vα contains exactly the sets with rank ≤ α. The stage Vα can also be described as the set of all sets with rank strictly less than α, regardless of whether α is zero, a successor, or a limit. This gives another recursive definition: Vα is the union of the power sets of all earlier stages. Plugging this back into the definition of rank yields a self-contained recursive formula: the rank of S is the union of (rank(z) + 1) for every z in S.

The first five stages V0 through V4 can be visualized, with an empty box for the empty set, a box containing an empty box for the set containing only the empty set, and so on. This sequence grows tetrationally. V5 has 2^16 = 65,536 elements; V6 has 2^65,536 elements, far more than the number of atoms in the observable universe; and for any natural number n, Vn+1 has 2↑↑n elements (using Knuth’s up-arrow notation). So after stage 5, the finite stages cannot be written out explicitly. Vω has the same cardinality as ω, and Vω+1 has the same cardinality as the set of real numbers.

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