Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T02:03:13.005Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Aczel, P., The type theoretic interpretation of constructive set theory, Logic colloquium '77, North Holland, 1978, pp. 55–66.Google Scholar
[2] Aczel, P. and Rathjen, M., Notes on constructive set theory, Technical Report 40, 2000/2001, Mittag–Leffler Institutes, Sweden, 2001.Google Scholar
[3] Awodey, S., An outline of algebraic set theory, in preparation, 2007.Google Scholar
[4] Awodey, S., Butz, C., Simpson, A.K., and Streicher, T., Relating first-order set theories, toposes and categories of classes, in preparation, 2007.Google Scholar
[5] Butz, C., Bernays–Gödel type theory, Journal of Pure and Applied Algebra, vol. 178 (2003), pp. 1–23.Google Scholar
[6] Fourman, M.P., Sheaf models for set theory, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 91–101.Google Scholar
[7] Friedman, H., Set theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105 (1977), pp. 1–28.Google Scholar
[8] Hayashi, S., On set theories in toposes, Logic Symposia, Hakone 1979, 1980, Lecture Notes in Mathematics, vol. 891, Springer, 1981, pp. 23–29.CrossRefGoogle Scholar
[9] Johnstone, P. T., Sketches of an elephant: a topos theory compendium, vol. 1–2, Oxford Logic Guides, no. 43–44, Oxford University Press, 2002.Google Scholar
[10] Joyal, A. and Moerdijk, I., Algebraic set theory, LMS Lecture Notes 220, Cambridge University Press, 1995.CrossRefGoogle Scholar
[11] Lambek, J. and Scott, P.J., Introduction to higher order categorical logic, Cambridge University Press, 1986.Google Scholar
[12] Lawvere, F. W. and Rosebrugh, R., Sets for mathematics, Cambridge University Press, 2003.CrossRefGoogle Scholar
[13] Lane, S. Mac and Moerdijk, I., Sheaves in geometry and logic: a first introduction to topos theory, Universitext, Springer Verlag, 1992.Google Scholar
[14] Mathias, A. R. D., The strength of Mac Lane set theory, Annals of Pure and Applied Logic, vol. 110 (2001), pp. 107–234.CrossRefGoogle Scholar
[15] Myhill, J., Constructive set theory, The Journal of Symbolic Logic, vol. 40 (1975), pp. 347–382.CrossRefGoogle Scholar
[16] Ščedrov, A., Intuitionistic set theory, Harvey Friedman's research on the foundations of mathematics, Elsevier Science Publishers, 1985, pp. 257–284.CrossRefGoogle Scholar
[17] Simpson, A. K., Elementary axioms for categories of classes, Proceedings of the 14th annual IEEE symposium on logic in computer science, 1999, pp. 77–85.Google Scholar
[18] Simpson, A. K. Computational adequacy for recursive types in models of intuitionistic set theory, Annals ofPure and Applied Logic, vol. 130 (2004), pp. 207–275.Google Scholar
[19] Simpson, A. K. Constructive set theories and their category-theoretic models. From sets and types to topology and analysis, Oxford University Press, 2005.Google Scholar
[20] Troelstra, A. S. and van Dalen, D., Constructivism in mathematics, vol. I–II, North Holland, Amsterdam, 1988.Google Scholar