Advertisement

Unique Existence and Computability in Constructive Reverse Mathematics

  • Hajime Ishihara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

Abstract

We introduce, and show the equivalences among, relativized versions of Brouwer’s fan theorem for detachable bars (FAN), weak König lemma with a uniqueness hypothesis (WKL!), and the longest path lemma with a uniqueness hypothesis (LPL!) in the spirit of constructive reverse mathematics. We prove that a computable version of minimum principle: if f is a real valued computable uniformly continuous function with at most one minimum on {0,1} N , then there exists a computable α in {0,1} N such that \(f(\alpha) = \inf f(\{0,1\}^\mathbf{N})\), is equivalent to some computably relativized version of FAN, WKL! and LPL!.

Keywords

unique existence computability Brouwer’s fan theorem weak König lemma constructive mathematics reverse mathematics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berger, J., Bridges, D., Schuster, P.: The fan theorem and unique existence of maxima. J. Symbolic Logic 71, 713–720 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, J., Ishihara, H.: Brouwer’s fan theorem and unique existence in constructive analysis. MLQ Math. Log. Q. 51, 360–364 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  4. 4.
    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bridges, D.: Recent progress in constructive approximation theory. In: Troelstra, A.S., van Dalen, D. (eds.) The L.E.J Brouwer Centenary Symposium, pp. 41–50. North-Holland, Amsterdam (1982)CrossRefGoogle Scholar
  6. 6.
    Bridges, D., Richman, F.: Varieties of Constructive Mathematics. Cambridge University Press, Cambridge (1987)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ishihara, H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36. 237–240 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ishihara, H.: Constructive reverse mathematics: compactness properties. In: Crosilla, L., Schuster, P. (eds.) From Sets and Types to Topology and Analysis, pp. 245–267. Oxford Univ. Press, Oxford (2005)CrossRefGoogle Scholar
  9. 9.
    Ishihara, H.: Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiæ, Cahier spécial 6, 43–59 (2006)CrossRefGoogle Scholar
  10. 10.
    Ishihara, H.: Weak König lemma implies Brouwer’s fan theorem: a direct proof. Notre Dame J. Formal Logic 47, 249–252 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Julian, W., Richman, F.: A uniformly continuous function on [0,1] that is everywhere different from its infimum. Pacific J. Math. 111, 333–340 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kleene, S.C.: Recursive functions and intuitionistic mathematics. In: Graves, L.M., Hille, E., Smith, P.A., Zariski, O. (eds.) Proceedings of the International Congress of Mathematicians, Amer. Math. Soc. Providence, pp. 679–685 (1952)Google Scholar
  13. 13.
    Kleene, S.C., Vesley, R.E.: The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions. North-Holland, Amsterdam (1965)zbMATHGoogle Scholar
  14. 14.
    Kohlenbach, U.: Effective moduli from ineffective uniqueness proofs: An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation. Ann. Pure Appl. Logic 64, 27–94 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kohlenbach, U., Oliva, P.: Proof mining: a systematic way of analysing proofs in mathematics. In: Proc. Steklov Inst. Math. vol. 242, pp. 136–164 (2003)zbMATHGoogle Scholar
  16. 16.
    Loeb, I.: Equivalents of the (weak) fan theorem. Ann. Pure Appl. Logic 132, 51–66 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schuster, P.: Unique solutions. MLQ Math. Log. Q. 52, 534–539 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schwichtenberg, H.: A direct proof of the equivalence between Brouwer’s fan theorem and König lemma with a uniqueness hypothesis. J. UCS 11, 2086–2095 (2005)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Simpson, S.G.: Subsystems of second order arithmetic. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, Vol. I An Introduction. North-Holland, Amsterdam (1988)zbMATHGoogle Scholar
  21. 21.
    Veldman, W.: Brouwer’s fan theorem as an axiom and as a contrast to Kleene’s alternative. Radboud University, Nijmegen (2005) (preprint)Google Scholar
  22. 22.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hajime Ishihara
    • 1
  1. 1.School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa 923-1292Japan

Personalised recommendations