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September 2007 Relating first-order set theories and elementary toposes
S. Awodey, C. Butz, A. Simpson, T. Streicher
Bull. Symbolic Logic 13(3): 340-358 (September 2007). DOI: 10.2178/bsl/1186666150

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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S. Awodey. C. Butz. A. Simpson. T. Streicher. "Relating first-order set theories and elementary toposes." Bull. Symbolic Logic 13 (3) 340 - 358, September 2007. https://doi.org/10.2178/bsl/1186666150

Information

Published: September 2007
First available in Project Euclid: 9 August 2007

zbMATH: 1152.03043
MathSciNet: MR2359910
Digital Object Identifier: 10.2178/bsl/1186666150

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.13 • No. 3 • September 2007
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