 
Summary: The Bulletin of Symbolic Logic
Volume 13, Number 3, Sept. 2007
RELATING FIRSTORDER SET THEORIES AND ELEMENTARY
TOPOSES
STEVE AWODEY, CARSTEN BUTZ, ALEX SIMPSON, AND THOMAS STREICHER
Abstract. We show how to interpret the language of firstorder set theory in an elementary
topos endowed with,as extrastructure, adirected structural system ofinclusions (dssi). As our
main result, weobtain acompleteaxiomatization oftheintuitionisticsettheoryvalidated byall
such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus
obtain a firstorder 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 ZermeloFraenkel set theory (IZF).
§1. Introduction. The notion of elementarytopos abstracts from the struc
ture of the category of sets. The abstraction is sufficiently general that
elementary toposes encompass a rich collection of other very different cat
egories, including categories that have arisen in fields as diverse as alge
braic geometry, algebraic topology, mathematical logic, and combinatorics.
Nonetheless, elementary toposes retain many of the essential features of the
category of sets. In particular, elementary toposes possess an internal logic,
