Feferman’s Forays into the Foundations of Category Theory

Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

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 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.

Keywords

Category theory Stratified set theory Zermelo-Fraenkel set theory Small/large distinction 

2010 Mathematics Subject Classification

03E70 18A15 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Philosophy, Linguistics, and Theory of ScienceUniversity of GothenburgGöteborgSweden

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