Pi function

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. 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.

History

The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed to von Neumann. The first publication of the von Neumann universe was by Ernst Zermelo in 1930.

Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory and Neumann's own set theory (which later developed into NBG set theory). In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays and Mendelson both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.

The notation V is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals. Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910. The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper by Gödel, although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.

The formula V = ⋃_αV_α is often considered to be a theorem, not a definition. Roitman states (without references) that the realization that the axiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.

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, {}:

  $V_{0}:=\{\}.$

• For any ordinal number β, let V_β+1 be the power set of V_β:

  $V_{\beta +1}:={\mathcal {P}}(V_{\beta }).$

• 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 $S\subseteq V_{\alpha }\,.$

The first five von Neumann stages V₀ to V₄ may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)

First 5 von Neumann stages

The set V₅ contains 65536 elements. The set V₆ contains 2⁶⁵⁵³⁶ elements, which very substantially exceeds the number of atoms in the known universe. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set V_ω has the same cardinality as ω. The set V_ω+1 has the same cardinality as the set of real numbers.

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 (ZFC) 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.

A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets" which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930. Such urelements are used extensively in model theory, particularly in Fraenkel-Mostowski models.

Philosophical perspectives

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See also

• Universe (mathematics)
• Constructible universe
• Grothendieck 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.