Feferman's Forays into the Foundations of Category Theory

TL;DR

This paper evaluates Feferman's set-theoretical system $S^*$ as a foundation for category theory, demonstrating its ability to handle unrestricted categories while maintaining ZFC-like properties, and improves the understanding of its consistency strength.

Contribution

It introduces an improved upper bound on the consistency strength of $S^*$ and shows how it can serve as a foundation for unrestricted category theory.

Findings

$S^*$ satisfies the unrestricted existence of key categories.

A recursive construction using strongly cantorian sets demonstrates $S^*$'s properties.

Provides motivation for NFU-based foundations in category theory.

Abstract

This paper is primarily concerned with assessing a set-theoretical system, $S^*$, for the foundations of category theory suggested by Solomon Feferman. $S^*$ is an extension of NFU, and may be seen as an attempt to accommodate unrestricted categories such as the category of all groups (without any small/large restrictions), while still obtaining the benefits of ZFC on part of the domain. A substantial part of the paper is devoted to establishing an improved upper bound on the consistency strength of $S^*$. The assessment of $S^*$ as a foundation of category theory is framed by the following general desiderata (R) and (S). (R) asks for the unrestricted existence of the category of all groups, the category of all categories, the category of all functors between two categories, etc., along with natural implementability of ordinary mathematics and category theory. (S) asks for a certain relative distinction between large and small sets, and the requirement that they both enjoy the full benefits of the $\mathrm{ZFC}$ axioms. $S^*$ satisfies (R) simply because it is an extension of NFU. By means of a recursive construction utilizing the notion of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this construction yields a lower bound on the consistency strength of $S^*$. We also exhibit a basic positive result for category theory internal to NFU that provides motivation for studying NFU-based foundations of category theory.