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Concept
Von Neumann universe
Summary
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. 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.
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 states "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:
The first five von Neumann stages V0 to V4 may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)
This sequence exhibits tetrational growth. The set V5 contains 216 = 65536 elements; the set V6 contains 265536 elements, which very substantially exceeds the number of atoms in the known universe; and for any natural n, the set Vn+1 contains 2 ↑↑ n elements using Knuth's up-arrow notation.
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