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

Published online by Cambridge University Press:  12 March 2014

Michael Rathjen
Michael Rathjen*
Affiliation:
Department of Mathematics, Ohio State University, Columbus. OH 43210., USA, E-mail: [email protected]
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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.

Keywords

Constructive set theoryrealizabilitymetamathematical property
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 4 , December 2005 , pp. 1233 - 1254
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Aczel, P., The type theoretic interpretation of constructive set theory, Logic Colloquium '77 (MacIntyre, A., Pacholski, L., and Paris, J., editors). North Holland, 1978. pp. 55–66.CrossRefGoogle Scholar
[2]Aczel, P., The type theoretic interpretation of constructive set theory: Choice principles, The L. E. J. Brouwer Centenary Symposium (Troelstra, A. S. and van Dalen, D., editors). North Holland, 1982, pp. 1–40.Google Scholar
[3]Aczel, P., The type theoretic interpretation of constructive set theory: Inductive definitions, Logic, Methodology and Philosophy of Science, VII (Marcus, R. B., Dorn, G. J. W., and Weingartner, P., editors). North Holland, 1986, pp. 17–49.Google Scholar
[4]Aczel, P. and Rathjen, M., 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.Google Scholar
[5]Barwise, J., Admissible Sets and Structures, Springer-Verlag, 1975.CrossRefGoogle Scholar
[6]Beeson, M., Continuity in intuilionistic set theories, Logic Colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North Holland, 1979.Google Scholar
[7]Beeson, M., Foundations of Constructive Mathematics, Springer-Verlag, 1985.CrossRefGoogle Scholar
[8]Crosilla, L. and Rathjen, M., Inaccessible set axioms may have little consistency strength, Annals of Pure and Applied Logic, vol. 115 (2002). pp. 33–70.CrossRefGoogle Scholar
[9]Feferman, S., A language and axioms for explicit mathematics, Algebra and Logic (Crossley, J. N., editor). Lecture Notes in Mathematics, vol. 450, Springer, 1975, pp. 87–139.CrossRefGoogle Scholar
[10]Feferman, S., Constructive theories of functions and classes, Logic Colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North Holland, 1979, pp. 159–224.Google Scholar
[11]Friedman, H., Some applications of Kleene's methodfor intuilionistic systems, Cambridge Summer School in Mathematical Logic (Mathias, A. and Rogers, H., editors). Lectures Notes in Mathematics, vol. 337, Springer, 1973, pp. 113–170.CrossRefGoogle Scholar
[12]Friedman, H., 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. 2877–2878.CrossRefGoogle Scholar
[13]Friedman, H., Set-theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105 (1977), pp. 868–870.CrossRefGoogle Scholar
[14]Friedman, H. and Ščedrov, S., Set existence property for intuilionistic theories with dependent choice, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 129–140.CrossRefGoogle Scholar
[15]Friedman, H. and Ščedrov, S., The lack of definable witnesses and provably recursive functions in intuitionistic set theory, Advances in Mathematics, vol. 57 (1985). pp. 1–13.CrossRefGoogle Scholar
[16]Kleene, S. C., On the interpretation of intuitionistic number theory, this Journal, vol. 10 (1945), pp. 109–124.Google Scholar
[17]Kleene, S. C., Disjunction and existence under implication in elementary intuitionistic formalisms, this JOurnal, vol. 27 (1962), pp. 11–18.Google Scholar
[18]Kleene, S. C., An addendum, this JOurnal, vol. 28 (1963), pp. 154–156.Google Scholar
[19]Kleene, S. C., Formalized recursive junctionals and formalized realizability, Memoirs of the American Mathematical Society, vol. 89 (1969).Google Scholar
[20]Kreisel, G. and Troelstra, A. S., Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1 (1970), pp. 229–387.CrossRefGoogle Scholar
[21]Lipton, J., Realizability, set theory and term extraction, The Curry-Howard Isomorphism, Cahiers du Centre de Logique de l'Universite Catholique de Louvain, vol. 8, 1995, pp. 257–364.Google Scholar
[22]McCarty, D. C., Realizability and recursive mathematics, Ph.D. thesis, Oxford University, 1984.Google Scholar
[23]McCarty, D. C., Realizability and recursive set theory, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 153–183.CrossRefGoogle Scholar
[24]Moschovakis, J. R., Disjunction and existence in formalized intuitionistic analysis, Sets, Models and Recursion Theory (Crossley, J. N., editor), North-Holland, 1967, pp. 309–331.CrossRefGoogle Scholar
[25]Myhill, J., Some properties of intuitionistic Zermelo-Fraenkel set theory, Cambridge Summer School in Mathematical Logic (Mathias, A. and Rogers, H., editors). Lecture Notes in Mathematics, vol. 337. Springer, 1973, pp. 206–231.CrossRefGoogle Scholar
[26]Myhill, J., Constructive set theory, this JOurnal, vol. 40 (1975), pp. 347–382.Google Scholar
[27]Rathjen, M., Realizability for constructive Zermelo-Fraenkel set theory, Logic Colloquium '03 (Väänänen, J. and Hansen, V. Stoltenberg, editors), to appear.Google Scholar
[28]Rathjen, M., The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347–385.Google Scholar
[29]Rathjen, M. and Tupailo, S., Characterizing the interpretation of set theory in Martin-Löf type theory, Annals of Pure and Applied Logic, to appear.Google Scholar
[30]Tharp, L., A quasi-intuitionistic set theory, this JOurnal, vol. 36 (1971), pp. 456–460.Google Scholar
[31]Troelstra, A. S., Realizability. Handbook of Proof Theory (Buss, S. R., editor), Elsevier, 1998, pp. 407–473.CrossRefGoogle Scholar
[32]Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, Volumes I, II, North Holland, 1988.Google Scholar