Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted 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 rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.[1] 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 a transfinite hierarchy, called the cumulative hierarchy, based on their rank.

Contents

Definition

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:

A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that defines "the set x is in Vα".

The class V is defined to be the union of all the V-stages:

V�:= \bigcup_{\alpha} V_\alpha.

An equivalent definition sets

V_\alpha�:= \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta)

for each ordinal α, where \mathcal{P} (X) \! is the powerset of X.

The rank of a set S is the smallest α such that S \subseteq V_\alpha \,.

V and set theory

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", and is a model of Zermelo set theory. If κ is an inaccessible cardinal, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and Vκ+1 is a model of Morse–Kelley set theory.

V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (for example is Aczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.

Philosophical perspectives

There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with 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.

See also

References

  1. ^ Mirimanoff 1917; Moore 1982, pp. 261-262; Rubin 1967, p. 214