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 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. 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.

Definition

The cumulative hierarchy is a collection of sets V_α indexed by the class of ordinal numbers, in particular, V_α is the set of all sets having ranks less than α. Thus there is one set V_α for each ordinal number α; V_α may be defined by transfinite recursion as follows:

• Let V₀ be the empty set, {}.
• For any ordinal number β, let V_β+1 be the power set of V_β.
• For any limit ordinal λ, let V_λ be the union of all the V-stages so far:

  $V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.$

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 V_α".

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 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", and is a model of Zermelo set theory. If κ is an inaccessible cardinal, then V_κ is a model of Zermelo-Fraenkel set theory itself, and V_κ+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 V_α 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

• Constructible universe
• Inaccessible cardinal
• S (Boolos 1989)
• John von Neumann

References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.