Relating first-order set theories and elementary toposes

Bulletin of Symbolic Logic 13 (3):340-358 (2007)
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Abstract

We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo—Fraenkel set theory (IZF)

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Steve Awodey
Carnegie Mellon University

References found in this work

Constructivism in Mathematics: An Introduction.A. S. Troelstra & Dirk Van Dalen - 1988 - Amsterdam: North Holland. Edited by D. van Dalen.
Constructive set theory.John Myhill - 1975 - Journal of Symbolic Logic 40 (3):347-382.
The strength of Mac Lane set theory.A. R. D. Mathias - 2001 - Annals of Pure and Applied Logic 110 (1-3):107-234.

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