Problems as Solutions

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

Abstract

If a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root. We validate this heuristic principle in Bishop–style constructive mathematics without countable choice, following Richman’s way of defining the completion of a metric space as the set of all locations.

MSC (2000): Primary 03F60; Secondary 03E25, 26E40, 54E50.

Keywords

Metric spaces completeness uniform continuity unique existence constructive mathematics countable choice 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczel, P.: The type theoretic interpretation of constructive set theory. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium ’77, pp. 55–66. North–Holland, Amsterdam (1978)CrossRefGoogle Scholar
  2. 2.
    Aczel, P., Crosilla, L., Ishihara, H., Palmgren, E., Schuster, P.: Binary refinement implies discrete exponentiation. Studia Logica 84, 361–368 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aczel, P., Rathjen, M.: Notes on Constructive Set Theory. Institut Mittag–Leffler Preprint No. 40 (2000/01)Google Scholar
  4. 4.
    Berger, J., Bridges, D., Schuster, P.: The fan theorem and unique existence of maxima. J. Symbolic Logic 71, 713–720 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berger, J., Ishihara, H.: Brouwer’s fan theorem and unique existence in constructive analysis. Math. Log. Quart. 51, 369–373 (2005)MathSciNetMATHGoogle Scholar
  6. 6.
    Bishop, E.: Foundations of Constructive Analysis. McGraw–Hill, New York (1967)MATHGoogle Scholar
  7. 7.
    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Bridges, D., Richman, F.: Varieties of Constructive Mathematics. Cambridge University Press, Cambridge (1987)CrossRefMATHGoogle Scholar
  10. 10.
    Burden, C.W., Mulvey, C.J.: Banach spaces in categories of sheaves. In: Fourman, M., Mulvey, C., Scott, D. (eds.) Applications of Sheaves. Proceedings, Durham, 1977. Lecture Notes in Math, vol. 753, pp. 169–196. Springer, Berlin and Heidelberg (1979)Google Scholar
  11. 11.
    Crosilla, L., Ishihara, H., Schuster, P.: On constructing completions. J. Symbolic Logic 70, 969–978 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ishihara, H.: Informal constructive reverse mathematics. Sūrikaisekikenkyūsho Kōkyūroku 1381, 108–117 (2004)Google Scholar
  13. 13.
    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
  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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kohlenbach, U., Oliva, P.: Proof mining: a systematic way of analysing proofs in mathematics. Proc. Steklov Inst. Math. 242(3), 136–164 (2003)MATHGoogle Scholar
  16. 16.
    Kreĭnovič, V.Ja.: Review of Constructive Functional Analysis. MR0521982 (82k:03094)Google Scholar
  17. 17.
    Lifshits, V.A.: Investigation of constructive functions by the method of fillings. J. Soviet Math. 1, 41–47 (1973)CrossRefMATHGoogle Scholar
  18. 18.
    Mulvey, C.J.: Banach spaces over a compact space. In: Herrlich, H., Preuss, G. (eds.) Categorical Topology. Proceedings, Berlin, 1978. Lecture Notes in Math, vol. 719, pp. 243–249. Springer, Berlin and Heidelberg (1979)Google Scholar
  19. 19.
    Palmgren, E.: A constructive and functorial embedding of locally compact metric spaces into locales. Department of Mathematics, Uppsala University, Report 25 (2006)Google Scholar
  20. 20.
    Rathjen, M.: Choice principles in constructive and classical set theories. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium ’02. Proceedings, Münster, 2002. Lect. Notes Logic 27, Assoc. Symbol. Logic, La Jolla pp. 299–326 (2006)Google Scholar
  21. 21.
    Richman, F.: Intuitionism as generalization. Philos. Math (3) 5, 124–128 (1990)MathSciNetMATHGoogle Scholar
  22. 22.
    Richman, F.: The fundamental theorem of algebra: a constructive development without choice. Pacific J. Math. 196, 213–230 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Richman, F.: Spreads and choice in constructive mathematics. Indag. Math. (N.S.) 13, 259–267 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Schuster, P.: Unique existence, approximate solutions, and countable choice. Theoret. Comput. Sci. 305, 433–455 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    Schuster, P.: Countable choice as a questionable uniformity principle. Philos. Math (3) 12, 106–134 (2004)MathSciNetMATHGoogle Scholar
  26. 26.
    Schuster, P.: Unique solutions. Math. Log. Quart. 52 (2006), pp. 534–539. Corrigendum: Math. Log. Quart. 53, 214 (2007)Google Scholar
  27. 27.
    Stolzenberg, G.: Sets as limits yellow. Typescript (1988)Google Scholar
  28. 28.
    Vickers, S.: Localic completion of generalised metric spaces. I. Theor. Appl. Categ. (15) (electronic) 14, 328–356 (2005)MATHGoogle Scholar
  29. 29.
    Vickers, S.: Localic completion of generalised metric spaces. II. Power locales. Preprint, School of Computer Science, University of Birmingham (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Peter Schuster
    • 1
  1. 1.Mathematisches Institut, Universität München, Theresienstr. 39, 80333 MünchenGermany

Personalised recommendations