Hostname: page-component-cc8bf7c57-fxdwj Total loading time: 0 Render date: 2024-12-11T22:15:23.944Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 (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