Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by 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 concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930.

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 the transfinite hierarchy , called the cumulative hierarchy, based on their rank.

Definition

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

Let be the empty set: For any ordinal number β, let be the power set of : For any limit ordinal λ, let be the union of all the V-stages so far: A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that states "(α is an ordinal and) the set x is in ".

The sets are called stages or ranks.

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

Rank of a set

The rank of a set S is the smallest α such that In other words, is the set of sets with rank ≤α. The stage can also be characterized as the set of sets with rank strictly less than α, regardless of whether α is 0, a successor ordinal, or a limit ordinal:

This gives an equivalent definition of by transfinite recursion.

Substituting the above definition of back into the definition of the rank of a set gives a self-contained recursive definition:

In other words,

Finite and low cardinality stages of the hierarchy

The first five von Neumann stages to 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.)

This sequence exhibits tetrational growth. The set contains 2 = 65536 elements; the set contains 2 elements, which very substantially exceeds the number of atoms in the observable universe; and for any natural , the set contains elements using Knuth's up-arrow notation. So the finite stages of the cumulative hierarchy cannot physically be written down explicitly after stage 5. The set has the same cardinality as ω. The set has the same cardinality as the set of real numbers.

Applications and interpretations Interpretation as the set-theoretical universe In the standard Zermelo–Fraenkel set theory, is simply the universe, i.e., the class of all sets. It is a proper class, and thus not "the set of all sets", even though each individual stage is a set, because the index ranges over the class of all ordinals, a proper class. The universality of also depends on the axiom of foundation (also known as the axiom of regularity). In non-well-founded set theories, the universe is larger than since the former also contains non-well-founded sets.

Often, is defined as the universe, and then the formula means "the universe of ZF sets is equal to the cumulative hierarchy"—not a definition, but a theorem equivalent to the axiom of regularity. Roitman states (without references) that the realization of this equivalence is due to von Neumann.. See page 79.

By the modern definition, also does not include urelements in the first stage, and thus only contains "pure sets". However, Zermelo's original construction of his transfinite recursive hierarchy in 1930 includes all urelements in ( in his notation), with the empty set considered a special case of an urelement.

Applications of V as models for set theories If ω is the set of natural numbers, then is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity., proves that: " is a model of all of the axioms of ZFC except infinity.", states: "The first really interesting axiom [of ZF set theory] is the Axiom of Infinity. If we drop it, then we can take as a model for ZF the set M of all finite sets which can be built up from ∅. [...] It is clear that M will be a model for the other axioms, since none of these lead out of the class of finite sets."

' is the universe of "ordinary mathematics", and is a model of Zermelo set theory (but not a model of ZF).. See page 96 for proof that is a Zermelo model. A simple argument in favour of the adequacy of is the observation that is adequate for the integers, while is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing the axiom of replacement to go outside .

If κ is an inaccessible cardinal, then is a model of Zermelo–Fraenkel set theory (ZFC) itself, and is a model of Morse–Kelley set theory., states and justifies that if κ is strongly inaccessible, then is a model of ZF. "It is clear that if A is an inaccessible cardinal, then the set of all sets of rank less than A is a model for ZF, since the only two troublesome axioms, Power Set and Replacement, do not lead out of the set of cardinals less than A.", proves that if κ is strongly inaccessible, then is a model of ZFC. (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)

Hilbert's paradox The von Neumann universe satisfies the following two properties:

for every set . for every subset .

Indeed, if , then for some ordinal . Any stage is a transitive set, hence every is already , and so every subset of is a subset of . Therefore, and . For unions of subsets, if , then for every , let be the smallest ordinal for which . Because by assumption is a set, we can form the limit . The stages are cumulative, and therefore again every is . Then every is also , and so and .

Hilbert's paradox implies that no set with the above properties exists .A. Kanamori, "Zermelo and Set Theory", p.490. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023. For suppose was a set. Then would be a subset of itself, and would belong to , and so would . But more generally, if , then . Hence, , which is impossible in models of ZFC such as itself.

Interestingly, is a subset of if, and only if, is a member of . Therefore, we can consider what happens if the union condition is replaced with . In this case, there are no known contradictions, and any Grothendieck universe satisfies the new pair of properties. However, whether Grothendieck universes exist is a question beyond ZFC.

The existential status of V

Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.See article On Formally Undecidable Propositions of Principia Mathematica and Related Systems and .

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers., . See also the English-language presentation of von Neumann's "general recursion theorem" by . The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.

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.. See page 279 for the assertion of the false attribution to von Neumann. See pages 270 and 281 for the attribution to Zermelo. The first publication of the von Neumann universe was by Ernst Zermelo in 1930.. See particularly pages 36–40.

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 von Neumann's own set theory (which later developed into NBG set theory).. See pages 745–752. 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. See pages 203–209. and Mendelson. See page 202. 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.. See pages VIII and XI. Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.. See page 229. 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. See page 88. 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.

Von Neumann universe

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