Von Neumann universe

In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets.

This may be defined by transfinite recursion as follows:

• Let V₀ be the empty set.
• For a an ordinal number, let V_a+ (where a+ is the successor ordinal of a) be the power set of V_a.
• For b a limit ordinal, let V_b be the union of all the V-stages so far:

<math> V_b := \bigcup_{a \in b} V_{a} \! <math>.

• Finally, let V be the union of all the V-stages:

<math> V := \bigcup_{a} V_{a} \! <math>.

If ω is the set of natural numbers, then V_ω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity. V_ω+ω is the universe of "ordinary mathematics", which is a model of Zermelo set theory. If k is an inaccessible cardinal, then V_k is a model of Zermelo-Fraenkel set theory itself.

Note that every individual stage V_a is a set, but their union V is a proper class. The sets in V are called hereditarily well-founded sets; the axiom of foundation demands that every set is well founded (and hence hereditarily well-founded. (Other axiom systems, omitting the axiom of regularity, or replacing it by a strong negation, such as Aczel's Anti-Foundation axiom, are possible, but rarely used.)

Given any set A, the smallest ordinal number i such that A belongs to V_i is the hereditary rank of A.