Relating first-order set theories and elementary toposes

Steve Awodey, Carsten Butz, Thomas Streicher, Alexander Simpson

Research output: Contribution to journalArticlepeer-review


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).
Original languageEnglish
Pages (from-to)340-358
Number of pages19
JournalBulletin of Symbolic Logic
Publication statusPublished - 2007


Dive into the research topics of 'Relating first-order set theories and elementary toposes'. Together they form a unique fingerprint.

Cite this