Journal of Symbolic Logic

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

Michael Rathjen

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Symbolic Logic Volume 70, Issue 4 (2005), 1233-1254.

Dates
First available in Project Euclid: 18 October 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1129642124

Digital Object Identifier
doi:10.2178/jsl/1129642124

Mathematical Reviews number (MathSciNet)
MR2194246

Zentralblatt MATH identifier
1100.03046

Subjects
Primary: 03F50: Metamathematics of constructive systems 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]

Keywords
Constructive set theory realizability metamathematical property

Citation

Rathjen, Michael. The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005), no. 4, 1233--1254. doi:10.2178/jsl/1129642124. http://projecteuclid.org/euclid.jsl/1129642124.


Export citation

References

  • 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. 55--66.
  • 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.
  • J. Barwise Admissible Sets and Structures, Springer-Verlag,1975.
  • M. Beeson Continuity in intuitionistic set theories, Logic Colloquium '78 (M. Boffa, D. van Dalen, and K. McAloon, editors), North Holland,1979.
  • L. Crosilla and M. Rathjen Inaccessible set axioms may have little consistency strength, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 33--70.
  • S. Feferman A language and axioms for explicit mathematics, Algebra and Logic (J. N. Crossley, editor), Lecture Notes in Mathematics, vol. 450, Springer,1975, pp. 87--139.
  • H. Friedman Some applications of Kleene's method for intuitionistic systems, Cambridge Summer School in Mathematical Logic (A. Mathias and H. Rogers, editors), Lectures Notes in Mathematics, vol. 337, Springer,1973, pp. 113--170.
  • H. Friedman and S. Ščedrov Set existence property for intuitionistic theories with dependent choice, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 129--140.
  • S. C. Kleene On the interpretation of intuitionistic number theory, Journal of Symbolic Logic, vol. 10 (1945), pp. 109--124.
  • G. Kreisel and A. S. Troelstra Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1 (1970), pp. 229--387.
  • J. Lipton 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.
  • D. C. McCarty Realizability and recursive mathematics, Ph.D. thesis, Oxford University,1984.
  • J. R. Moschovakis Disjunction and existence in formalized intuitionistic analysis, Sets, Models and Recursion Theory (J. N. Crossley, editor), North-Holland,1967, pp. 309--331.
  • 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. 206--231.
  • M. Rathjen Realizability for constructive Zermelo-Fraenkel set theory, Logic Colloquium '03 (J. Väänänen and V. Stoltenberg Hansen, editors), to appear.
  • M. Rathjen and S. Tupailo Characterizing the interpretation of set theory in Martin-Löf type theory, Annals of Pure and Applied Logic, to appear.
  • L. Tharp A quasi-intuitionistic set theory, Journal of Symbolic Logic, vol. 36 (1971), pp. 456--460.
  • A. S. Troelstra Realizability, Handbook of Proof Theory (S. R. Buss, editor), Elsevier,1998, pp. 407--473.
  • A. S. Troelstra and D. van Dalen Constructivism in Mathematics, Volumes I, II, North Holland,1988.