Von Neumann universe

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.

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 states "the set x is in Vα".

The stage Vα 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 Vα by transfinite recursion.

[4] A simple argument in favour of the adequacy of Vω+ω is the observation that Vω+1 is adequate for the integers, while Vω+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 Vω+ω.

These non-well-founded set theories are not commonly employed, but are still possible to study.

Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930.

[8] The von Neumann universe satisfies the following two properties: Indeed, if

Hilbert's paradox implies that no set with the above properties exists .

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.

[10][11] 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.

[12] 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.

[13] 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.

[14] The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.

[7] 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.

[15] The first publication of the von Neumann universe was by Ernst Zermelo in 1930.

[17] 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[10] and Mendelson[11] 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.

[18] Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.

[19] 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[20] explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper by Gödel,[21] although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.

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.