Bulletin of Symbolic Logic

Relating first-order set theories and elementary toposes

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).

Article information

Source
Bull. Symbolic Logic Volume 13, Issue 3 (2007), 340-358.

Dates
First available in Project Euclid: 9 August 2007

http://projecteuclid.org/euclid.bsl/1186666150

Digital Object Identifier
doi:10.2178/bsl/1186666150

Mathematical Reviews number (MathSciNet)
MR2359910

Zentralblatt MATH identifier
1152.03043

Citation

Awodey, S.; Butz, C.; Simpson, A.; Streicher, T. Relating first-order set theories and elementary toposes. Bulletin of Symbolic Logic 13 (2007), no. 3, 340--358. doi:10.2178/bsl/1186666150. http://projecteuclid.org/euclid.bsl/1186666150.