Publication

Feferman’s forays into the foundations of category theory

Enayat, Ali
; 
Gorbow, Paul
; 
McKenzie, Zachiri
Advisors
Editors
Jäger, Gerhard
Sieg, Wilfried
Other Contributors
Affiliation
University of Gothenburg
EPub Date
Publication Date
2018-04-02
Submitted Date
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Mathematics
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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.
Citation
Enayat, A., Gorbow, P., & McKenzie, Z. (2018). Feferman’s forays into the foundations of category theory. In G. Jäger & W. Sieg (Eds.), Feferman on Foundations (pp. 315-346). Springer. https://doi.org/10.1007/978-3-319-63334-3_12
Publisher
Springer
Journal
Research Unit
URI
http://hdl.handle.net/10034/629351
DOI
10.1007/978-3-319-63334-3_12
PubMed ID
PubMed Central ID
Type
Book chapter
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Description
This book chapter is not available on ChesterRep
Series/Report no.
Outstanding Contributions to Logic; vol 13
ISSN
EISSN
ISBN
9783319633329
ISMN
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Sponsors
Unfunded
Additional Links
https://link.springer.com/chapter/10.1007/978-3-319-63334-3_12