On the Constructive Dedekind Reals: Extended Abstract

  • Robert S. Lubarsky
  • Michael Rathjen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4514)


In order to built the collection of Cauchy reals as a set in constructive set theory, the only Power Set-like principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.


Transition Function Ground Model Kripke Model Proper Class Elementary Embedding 
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  1. 1.
    Aczel, P.: The type theoretic interpretation of constructive set theory. In: MacIntypre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium ’77, pp. 55–66. North-Holland, Amsterdam (1978)Google Scholar
  2. 2.
    Aczel, P.: The type theoretic interpretation of constructive set theory: choice principles. In: Troelstra, A.S., van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, pp. 1–40. North-Holland, Amsterdam (1982)Google Scholar
  3. 3.
    Aczel, P.: The type theoretic interpretation of constructive set theory: inductive definitions. In: Marcus, R.B., et al. (eds.) Logic, Methodology and Philosophy of Science VII, pp. 17–49. North-Holland, Amsterdam (1986)Google Scholar
  4. 4.
    Aczel, P., Rathjen, M.: Notes on constructive set theory. Technical Report 40, 2000/2001, Mittag-Leffler Institute, Sweden (2001)Google Scholar
  5. 5.
    Crosilla, L., Ishihara, H., Schuster, P.: On constructing completions. The Journal of Symbolic Logic 70, 969–978 (2005)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Fourman, M.P., Hyland, J.M.E.: Sheaf models for analysis. In: Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.) Applications of sheaves, pp. 280–301. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  7. 7.
    Friedman, H., Scedrov, A.: The lack of definable witnesses and provably recursive functions in intuitionistic set theories. In: Advances in Math., pp. 1–13 (1985)Google Scholar
  8. 8.
    Lubarsky, R.: Independence results around Constructive ZF. Annals of Pure and Applied Logic 132, 209–225 (2005)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Lubarsky, R.: CZF + full Separation is equivalent to second order arithmetic. Annals of Pure and Applied Logic 141, 29–34 (2006)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Martin-Löf, P.: An intuitionistic theory of types: predicative part. In: Rose, H.E., Sheperdson, J. (eds.) Logic Colloquium ’73, pp. 73–118. North-Holland, Amsterdam (1975)Google Scholar
  11. 11.
    Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis, Naples (1984)MATHGoogle Scholar
  12. 12.
    Myhill, J.: Constructive set theory. Journal of Symbolic Logic, 347–382 (1975)Google Scholar
  13. 13.
    Rathjen, M.: The strength of some Martin–Löf type theories. Archive for Mathematical Logic, 347–385 (1994)Google Scholar
  14. 14.
    Rathjen, M.: The higher infinite in proof theory. In: Makowsky, J.A., Ravve, E.V. (eds.) Logic Colloquium ’95. Springer Lecture Notes in Logic, vol. 11, pp. 275–304. Springer, New York (1998)Google Scholar
  15. 15.
    Simpson, A.: Constructive set theories and their category-theoretic models. In: Crosilla, L., Schuster, P. (eds.) From Sets and Types to Topology and Analysis. Oxford Logic Guides, Oxford University Press, Oxford (forthcoming)Google Scholar
  16. 16.
    Streicher, T.: Realizability Models for IZF and CZF + ¬ Pow via the Aczel Construction. Personal communicationGoogle Scholar
  17. 17.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, Volumes I, II. North-Holland, Amsterdam (1988)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Robert S. Lubarsky
    • 1
  • Michael Rathjen
    • 2
  1. 1.Department of Mathematical Sciences, Florida Atlantic University 
  2. 2.Department of Pure Mathematics, University of Leeds 

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