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The disjunction and related properties for constructive Zermelo-Fraenkel set theory

  • Michael Rathjen (a1)
Abstract
Abstract

This paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.

As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.

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[1]P. Aczel , The type theoretic interpretation of constructive set theory, Logic Colloquium '77 (A. MacIntyre , L. Pacholski , and J. Paris , editors). North Holland, 1978. pp. 5566.

[2]P. Aczel , The type theoretic interpretation of constructive set theory: Choice principles, The L. E. J. Brouwer Centenary Symposium (A. S. Troelstra and D. van Dalen , editors). North Holland, 1982, pp. 140.

[4]P. Aczel and M. Rathjen , Notes on constructive set theory, Technical report, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences. Stockholm, 2001, TR-40, available at http://www.ml.kva.se/preprints/archive2000-2001.php.

[5]J. Barwise , Admissible Sets and Structures, Springer-Verlag, 1975.

[7]M. Beeson , Foundations of Constructive Mathematics, Springer-Verlag, 1985.

[8]L. Crosilla and M. Rathjen , Inaccessible set axioms may have little consistency strength, Annals of Pure and Applied Logic, vol. 115 (2002). pp. 3370.

[11]H. Friedman , Some applications of Kleene's methodfor intuilionistic systems, Cambridge Summer School in Mathematical Logic (A. Mathias and H. Rogers , editors). Lectures Notes in Mathematics, vol. 337, Springer, 1973, pp. 113170.

[12]H. Friedman , The disjunction property implies the numerical existence properly, Proceedings of the National Academy of Sciences of the United States of America, vol. 72 (1975), pp. 28772878.

[13]H. Friedman , Set-theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105 (1977), pp. 868870.

[14]H. Friedman and S. Ščedrov , Set existence property for intuilionistic theories with dependent choice, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 129140.

[15]H. Friedman and S. Ščedrov , The lack of definable witnesses and provably recursive functions in intuitionistic set theory, Advances in Mathematics, vol. 57 (1985). pp. 113.

[20]G. Kreisel and A. S. Troelstra , Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1 (1970), pp. 229387.

[23]D. C. McCarty , Realizability and recursive set theory, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 153183.

[24]J. R. Moschovakis , Disjunction and existence in formalized intuitionistic analysis, Sets, Models and Recursion Theory (J. N. Crossley , editor), North-Holland, 1967, pp. 309331.

[25]J. Myhill , Some properties of intuitionistic Zermelo-Fraenkel set theory, Cambridge Summer School in Mathematical Logic (A. Mathias and H. Rogers , editors). Lecture Notes in Mathematics, vol. 337. Springer, 1973, pp. 206231.

[31]A. S. Troelstra , Realizability. Handbook of Proof Theory (S. R. Buss , editor), Elsevier, 1998, pp. 407473.

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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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