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, 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 V_α, 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:

A crucial fact about this definition is that there is a single formula φ in the language of ZFC that states " the set x is in V_α".
The sets V_α 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 V_α 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 V_α by transfinite recursion.
Substituting the above definition of V_α 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 V₀ to V₄ may be visualized as follows.
This sequence exhibits tetrational growth. The set V₅ contains 2¹⁶ = 65536 elements; the set V₆ contains 2⁶⁵⁵³⁶ elements, which very substantially exceeds the number of atoms in the observable universe; and for any natural, the set V_n+1 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 V_ω has the same cardinality as ω. The set V_ω+1 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. 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 that the realization of this equivalence is due to von Neumann.
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 , 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 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. A simple argument in favour of the adequacy of V_ω+ω is the observation that V_ω+1 is adequate for the integers, while V_ω+2 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 V_ω+ω.
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.

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. 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.
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. 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 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 von Neumann's own 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 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 to a 1940 paper by Gödel, although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.

Philosophical perspectives

There are two approaches to understanding the relationship of the von Neumann universe V to ZFC. Roughly, formalists will tend to view V as something that flows from the ZFC axioms. 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, but does not necessarily describe objects with real existence.