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Algebraic set theory and the effective topos

  • Claire Kouwenhoven-Gentil (a1) and Jaap van Oosten (a2)
Abstract

Following the book Algebraic Set Theory from André Joyal and Ieke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).

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[1]Awodey, S., Butz, C., Simpson, A. K., and Streicher, T., Relating topos theory and set theory via categories of classes, Working draft, available at http://www.phil.emu.edu/projects/ast/index.html, 2003.
[2]Beeson, M. J., Foundations of constructive mathematics, Springer-Verlag, 1985.
[3]Friedman, H. M., Some applications of Kleene's methods for intuitionistic systems, Cambridge summer school in mathematical logic (Mathias, A. R. D. and Rogers, H., editors), Springer-Verlag, 1973, pp. 113170.
[4]Grayson, R. J., Heyting-valued models for intuitionistic set theory, Application of sheaves (Fourman, M., Mulvey, C., and Scott, D. S., editors), Lecture Notes in Mathematics, vol. 743, Springer, Berlin, 1979, pp. 402414.
[5]Hofmann, M., van Oosten, J., and Streicher, T., Well-foundedness in readability, submitted; available at http://www.math,uu.nl/people/jvoosten/realizability/welfreal.ps.gz, 2005.
[6]Hyland, J. M. E., The effective topos, The L. E. J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North Holland Publishing Company, 1982, pp. 165216.
[7]Joyal, A. and Moerdijk, I., A categorical theory of cumulative hierarchies of sets, Comptes Rendus Mathematiques de l'Académic des Sciences. La Société Royale du Canada, vol. 12 (1990), pp. 253260.
[8]Joyal, A., Algebraic set theory, London Mathematical Society Lecture Note Series, vol. 220, Cambridge University Press, Cambridge, 1995.
[9]Mccarty, D. C., Realizability and recursive mathematics, Technical Report CMU–CS–84–131, Department of Computer Science, Carnegie-Mellon University, 1984, Report version of the author's PhD thesis, Oxford University 1983.
[10]Moerduk, Ieke and Palmgren, Erik, Wellfounded trees in categories, Annals of Pure and Applied Logic, vol. 104 (2000), pp. 189218.
[11]Moerduk, Ieke, Type theories, toposes and constructive set theory: predicative aspects of AST, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 155201.
[12]van Oosten, Jaap, Axiomatizing higher-order Kleene realizability, Annals of Pure and Applied Logic, vol. 70 (1994), pp. 87111.
[13]Robinson, E. P. and Rosolini, G., Colimit completions and the effective topos, this Journal, vol. 55 (1990), pp. 678699.
[14]Simpson, A. K., Elementary axioms for categories of classes, Proceedings of the 14th annual IEEE symposium on logic in computer science, 1999, pp. 7785.
[15]Troelstra, A. S. (editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics 344, Springer, 1973, With contributions by Troelstra, A. S., Smoryński, C. A., Zucker, J. I. and Howard, W. A..
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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