The Logical Strength of the Uniform Continuity Theorem

  • Josef Berger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

We introduce a notion of complexity for sets of finite binary sequences such that the corresponding fan theorem is constructively equivalent to the uniform continuity theorem. This settles an open question.

Keywords

Constructive reverse mathematics uniform continuity theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczel, P., Rathjen, M.: Notes on constructive set theory. Technical Report 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences (2001)Google Scholar
  2. 2.
    Berger, J.: The fan theorem and uniform continuity. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 18–22. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)MATHGoogle Scholar
  4. 4.
    Bishop, E., Bridges, D.: Constructive Analysis. In: Grundlehren der mathematischen Wissenschaften, vol. 279. Springer, Heidelberg (1985)Google Scholar
  5. 5.
    Bridges, D., Richman, F.: Varieties of Constructive Mathematics. London Mathematical Society Lecture Note Series, vol. 97. Cambridge University Press, Cambridge (1987)CrossRefMATHGoogle Scholar
  6. 6.
    Ishihara, H.: Constructive Reverse Mathematics: Compactness Properties. In: Crosilla, L., Schuster, P. (eds.) From Sets and Types to Topology and Analysis. Oxford Logic Guides, vol. 48, pp. 245–267. Oxford University Press, Oxford (2005)CrossRefGoogle Scholar
  7. 7.
    Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiae (to appear)Google Scholar
  8. 8.
    Loeb, I.: Equivalents of the (Weak) Fan Theorem. Annals of Pure and Applied Logic 132(1), 51–66 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. Springer, Heidelberg (1973)MATHGoogle Scholar
  10. 10.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics. An Introduction. In: Studies in Logic and the Foundation of Mathematics, vol. I, 121. North-Holland, Amsterdam (1988)Google Scholar
  11. 11.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics. An Introduction. In: Studies in Logic and the Foundation of Mathematics, vol. II, 123. North-Holland, Amsterdam (1988)Google Scholar
  12. 12.
    Veldman, W.: Brouwer’s fan theorem as an axiom and as a contrast to Kleene’s alternative. Radboud University, Nijmegen (preprint, 2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Josef Berger
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

Personalised recommendations