MacLane's Problem on Foundations of Category Theory and Local Theory of Sets
by
Valeri Zakharov
Moscow State University
Coauthors: P.V. Andreev, E.I. Bunina, A.V. Mikhalev

The crises arisen in naive set theory in the beginning of the 20th century brought to the origin of some strict axiomatic constructions of theories of mathematical totalities, for example, the theory of sets in Zermelo-Fraenkel's axiomatics (ZF) and the theory of classes and sets in von Neumann-Bernays-Gödel's axiomatics (NBG). These theories eliminated all the known paradoxes of naive set theory and gave the opportunity to include within the framework of these theories almost all mathematical objects and constructions existed at that moment. But in 1945 the new mathematical notion of a category was introduced by Eilenberg and MacLane. This notion did not go not only within the framework of the theory of sets ZF but even within the framework of the theory NBG. By this reason in 1959 in the paper [1] MacLane raised the general problem of constructing a new and more flexible axiomatic set theory which could serve as an adequate logical foundation for all naive category theory.

From that time some set theories were offered in the capacity of foundations for category theory. However they either did not serve all category theory, or were too strong for needs of category theory, or were too complicated to use in practice. In 2000 V.K. Zakharov and A.V. Mikhalev offered in the capacity of more adequate foundation of category theory a local theory of sets LTS (see announcements in [2] and [3]). The main idea of the LTS, justifying the word ``local'' in its title, consists in taking a universe of classes and sets NBG and duplicating it in copies to get some hierarchy of universes with the following properties:

1) each class belongs as a set to some universal class, which is a usual NBG-universe;

2) all subclasses of a given universe are the sets of each larger universe;

3) there exists the least universe ( \equiv infra-universe), which is contained in all other universes.

Thus a notion of a big totality becomes relative: totalities that are ``big'' in one universe become ``small'' in any larger universe.

After presenting a list of axioms and axiom schemes of the LTS we formalize in the framework of local universal classes of the LTS such the key category constructions as ``a category of categories'' and ``a category of functors''. This shows that the LTS serves all naive category theory.

To compare the LTS with the theory ZF we construct in the LTS classes of von Neumann and with their help we prove that the totality of all universal classes in the LTS is well-ordered. This allows us to construct a model of the theory ZF with the additional axiom IC (there exists a strongly inaccessible cardinal) in the LTS. Also we construct a model of the LTS in the theory ZF with the additional axiom ISIC (there exists an infinite set of strongly inaccessible cardinals). Finally, we construct a model of the theory ZF+ISIC in the LTS with the additional axiom ICU (there exists an infinite class of universes). This implies the following chain of relative consistences: cons (LTS+ICU) ===> cons(ZF+ISIC) ===> cons(LTS) ===> cons (ZF+IC). We prove also that the axiom ICU is independent of all other axioms of the LTS.

To show that the totality of all classes in the LTS can not serve in the capacity of an adequate receptacle of categories, functors, natural transformations, and so on we prove that if the LTS is consistent, then the axiom scheme of replacement is not deducible in the LTS.

References

[1] MacLane S. Locally small categories and the foundations of set theory // Symposium. Warsaw. 1959. P.25-43

[2] Zakharov V.K., Mikhalev A.V. Local theory of classes and sets as the foundation of the category theory // Math. methods and applications. Translations of the 9th mathematical lecturing of MGSU. Moscow: MGSU. 2002. P.91-94.

[3] Andreev P.V., Bunina E.I., Zakharov V.K., Mikhalev A.V. Foundations of category theory in the framework of local theory of sets // Abstracts of reports of International algebraic conference. Saint-Petersburg: PDMI. 2002. P.10-11.

Date received: February 24, 2003