Von Neumann universe

From Academic Kids

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:

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

Note that every individual stage Va 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 Vi is the hereditary rank of A.

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