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The Bulletin of Symbolic Logic Volume 13, Number 3, Sept. 2007
 

Summary: The Bulletin of Symbolic Logic
Volume 13, Number 3, Sept. 2007
RELATING FIRST-ORDER SET THEORIES AND ELEMENTARY
TOPOSES
STEVE AWODEY, CARSTEN BUTZ, ALEX SIMPSON, AND THOMAS STREICHER
Abstract. We show how to interpret the language of first-order 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 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 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,

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics