Skip to main content
    • Aa
    • Aa

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

  • Michael Rathjen (a1)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *