ON THE EQUICONSISTENCY OF ZFC AND ETCS WITH REPLACEMENT

Abstract

Discussions of the elementary theory of the category of sets (ETCS) often take for granted its ’equivalence’ with a form of conventional axiomatic set theory. The persuasiveness of such evocations of ’equivalence’ are complicated by their frequent omission of an axiom schema of replacement, even as their attendant expositions claim that the inclusion of replacement is generally unproblematic. Few sources test this assertion. In this expository paper, we articulate an axiom schema of replacement, R, within a categorical setting and prove the equiconsistency of ETCS + R and ZFC.