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.
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Notes
- 1.
Feferman [9, 13] has put forward poignant criticisms of the general case of using category theory as an autonomous foundation for mathematics. Moreover, he suggested that a theory of operations and collections should also be pursued as a viable alternative platform for category theory; e.g. systems of Explicit Mathematics [8] and Operational Set Theory [12].
- 2.
This equiconsistency result is due to Robert Solovay, whose 1995-proof is unpublished. The first-named-author used a different proof in [3] to establish a refinement of Solovay’s equiconsistency result.
- 3.
\(\mathrm {KM} + \Pi ^1_\infty \)-AC is bi-interpretable with \(T := \mathrm {ZFC} - \text {Power Set} + \exists \kappa \) “\(\kappa \) is an inaccessible cardinal, and \(\forall x \; |x| \le \kappa \)”, where \(\Pi ^1_\infty \)-AC is the schema of Choice whose instances are of the form \(\forall s \, \exists X \, \phi (s, X) \rightarrow \exists Y \, \forall s \, \phi (s, (Y)_s)\), where \(\phi (s, X)\) is a formula of class theory with set variable s and class variable Y, and \((Y)_s\) is the “s-th slice of Y”, i.e., \((Y)_s = \{x \mid \langle s, x \rangle \in Y\}\). This bi-interpretability was first noted by Mostowski; a modern account is given in a recent paper of Antos and Friedman [1, 2], where \(\mathrm {KM} + \Pi ^1_\infty \)-AC is referred to as \(\mathrm {MK}^*\), and T is referred to as \(\mathrm {SetMK}^*\).
- 4.
- 5.
The axioms of Mac Lane set theory are specified in Sect. 6.
- 6.
The proof given, for the separation and the weakened replacement schemata, quantifies over formulas in the meta-theory. So what is proved is actually a theorem schema about \(\bar{\bar{V}}\), where \(\bar{\bar{V}}\) is considered externally as a submodel of a model of \(S^*\). However, Roland Hinnion’s development of the semantics of first-order logic in [16] applies to NFU, and provides a means to internalize this proof to \(S^*\).
- 7.
Mathias omits powerset and transitive containment from his axiomatisation of \(\mathrm {KPR}\) in [22], but both of these axioms follow from the existence of \(V_\alpha \) for every ordinal \(\alpha \).
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Enayat, A., Gorbow, P., McKenzie, Z. (2017). Feferman’s Forays into the Foundations of Category Theory. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_12
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