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Relating First-Order Set Theories and Elementary Toposes

Published online by Cambridge University Press:  15 January 2014

Steve Awodey ,
Carsten Butz ,
Alex Simpson  and
Thomas Streicher
Steve Awodey
Affiliation:
Department of Philosophy, CMU Pittsburgh, USA, E-mail: awodey@cmu.edu
Carsten Butz
Affiliation:
IT, University of Copenhagen, Denmark, E-mail: butz@it-c.dk
Alex Simpson
Affiliation:
LFCS, School of Informatics, University of Edinburgh, UK, E-mail: Alex.Simpson@ed.ac.uk
Thomas Streicher
Affiliation:
Fachbereich Mathematik, Tu Darmstadt, Germany, E-mail: streicher@mathematik.tu-darmstadt.de
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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).

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 3 , September 2007 , pp. 340 - 358
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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