Von Neumann universe

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In 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.

Philosophical perspectives

There are two distinct approaches to understanding the relationship of the von Neumann universe V to ZFC (and many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.