Universal Sets, Tarski Sets, and Inaccessible von Neumann Sets
by
Valeri Zakharov
Moscow State University
Coauthors: Elena Bunina

The crises arisen in naive set theory in the beginning of the 20th century brought to the origin of some strict axiomatic theories. The most widely used of them are the theory of sets in Zermelo-Fraenkel's axiomatics (ZF) and the theory of classes ans sets in Neumann-Bernays-Gödel's axiomatics (NBG). These axiomatic theories eliminated all the known paradoxes of naive set theory and gave the opportunity to unclude within the framework of them almost all mathematical objects and constructions existed at that time. But in 1945 the new mathematical notion of a category was introduced by Eilenberg and MacLane [1]. From the very beginning of its origin category theory encountered with that unpleasant circumstance that it did not go within the framework of the theories ZF and NBG or any other set theory known at this time, because category theory contains some constructions which are not admissible in all these set theories.

Way out of this situation was proposed by logicians shortly after that. Developing the ideas of Tarski, they proposed to strengthen the theory ZF by additional axioms on existence of strongly inaccessible cardinals, because it was already known then that the von Neumann sets V_\varkappa for strongly inaccessible cardinals \varkappa ( º inaccessible von Neumann sets) are inner models of these theories and therefore are suitable for working with categories. If for some categorical constructions the set V_\varkappa is not sufficient, then we can go to some inaccessible von Neumann set V_l such that V_\varkappa Î V_l for some l > \varkappa. The change property A Ì V_\varkappaÞ A Î V_l means that sets which are big in V_\varkappa, become small in V_l. Therefore constructions over big sets from V_\varkappa which are impossible in V_\varkappa become possible in V_l.

This idea turned out to be rather fruitful, but specialists in category theory (Ehresmann, Dedecker, Sonner and Grothendieck) noticed that for working with categories it is not necessary to take into consideration the construction of inaccessible von Neumann sets V_\varkappa, but it is sufficient to choose some their properties which are required for defining all categorical notions. In such a way the notion of a universal set (of Ehresmann-Dedecker-Sonner-Grotendieck) U appeared (see [2]). This set has the following properties: 1) X Î UÞ X Ì U; 2) X Î UÞ P(X), ÈX Î U; 3) X, Y Î UÞ XÈY, { X, Y}, áX, Yñ, X×Y Î U; 4) X Î UÙ(F Î U^X)Þ rng F Î U; 5) w Î U (w º { 0, 1, 2, ...} is the set of all finite ordinal numbers).

For working with categories it is sufficient to strengthen the theory ZF by the axiom of universality, which postulates that every set is an element of some universal set. Within the framework of a universal set U we can define such notions as U-category, U-functor, U-natural transformation, and so on. If we need, we can move up to universal sets V such that U Î V.

Every inaccessible von Neumann set V_\varkappa is universal. But the question "is the axiomatic notion of a universal set more wide than the constructive notion of an inaccessible von Neumann set?" remained open.

We give an answer to this question: there are no universal sets exept inaccessible von Neumann sets V_\varkappa; this result was obtained in joint authorship with P.V. Andreev).

Thus the notion of a universal set gives an axiomatic description of inaccessible von Neumann sets.

A representation of universal sets in the form V_\varkappa implies that the class of all universal sets is well-ordered with respect to the order by inclusion  Ì , which is equal to the order =È Î .

Another properties of inaccessible von Neumann sets V_\varkappa were extracted by Tarski [3] in 1938. He introduced the notion of a Tarski set T, which is defined by the following properties: 1) X Î TÞ X Ì T; 2) X Î TÞ P(X) Î T; 3) ((X Ì T)Ù"f (f Î T^XÞ f[X] ¹ T))Þ X Î T.

For axiomatic constructing of strongly inaccessible cardinal numbers Tarski introduced into the theory ZF the Tarski axiom, which postulates that every set is an element of some Tarski set.

Every inaccessible von Neumann set V_\varkappa is a Tarski set. Therefore the following question remained open: is the axiomatic notion of a Tarski set more wide than the constructive notion of an inaccessible von Neumann set? Besides, the question "what correlation is between Tarski sets and universal sets?" also remained open.

We give answers to both of these questions: the notions of an inaccessible von Neumann set, a Tarski set of power greater than w, and a universal set are equivalent.

References.
[1] Eilenberg S., MacLane S. General theory of natural equivalences // Tranc. Amer. Math. Soc. 1945. V.58. P.231-294.
[2] Sonner J. The formal definition of categories // Math. Zeit. 1962. B.80. S.163-176.
[3] Tarski A. Über unerreichbare Kardinalzahlen // Fund. Math. 1938. V.30. P.68-89.

Date received: February 20, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cajy-02.