Brouwer’s Approximate Fixed-Point Theorem is Equivalent to Brouwer’s Fan Theorem

  • Wim Veldman
Part of the Synthese Library book series (SYLI, volume 341)


In a weak system for intuitionistic analysis, one may prove, using the Fan Theorem as an additional axiom, that, for every continuous function ø from the unit square U to itself, for every positive rational e, there exists x in U such that |ø(x) − x| < e. Conversely, if this statement is taken as an additional axiom, the Fan Theorem follows.


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  1. 1.
    J. Berger and H. Ishihara, Brouwer’s fan theorem and unique existence in constructive analysis, Math. Logic Quart. 51(2005), pp. 360–364.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L.E.J. Brouwer, Über Abbildungen von Mannigfaltigkeiten, Mathematische Annalen 71(1911), pp. 97–115, also in [7], pp. 454–472.CrossRefMathSciNetGoogle Scholar
  3. 3.
    L.E.J. Brouwer, Beweis dass jede volle Funktion gleichmässig stetig ist, Koninklijke Nederlandse Akademie van Wetenschappen Verslagen 27(1924), pp. 189–193, also in: [6], pp. 286–290.Google Scholar
  4. 4.
    L.E.J. Brouwer, An intuitionistic correction of the fixed-point theorem on the sphere, Proc. Roy. Soc. London , Ser. A 213(1952), pp. 1–2, also in [6], pp. 506–507.Google Scholar
  5. 5.
    L.E.J. Brouwer, Door klassieke theorema’s gesignaleerde pinkernen die onvindbaar zijn, Indag. math. 14(1952), pp. 443–445, translation: Fixed cores which cannot be found, though they are claimed to exist by classical theorems, in [6], pp. 519–521.zbMATHMathSciNetGoogle Scholar
  6. 6.
    L.E.J. Brouwer, Collected Works, Vol. I: Philosophy and Foundations of Mathematics, ed. A. Heyting, North Holland Publ. Co., Amsterdam, 1975.Google Scholar
  7. 7.
    L.E.J. Brouwer, Collected Works, Vol. II: Geometry, Analysis, Topology and Mechanics, ed. H. Freudenthal, North Holland Publ. Co., Amsterdam, 1976.Google Scholar
  8. 8.
    R. Courant, H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, Oxford, 1941, new edition, revised by I. Stewart, 1996.zbMATHGoogle Scholar
  9. 9.
    S.C. Kleene, R.E. Vesley, The Foundations of Intuitionistic Mathematics, Especially in Relation to the Theory of Recursive Functions, Amsterdam, North-Holland Publ. Co., 1965.Google Scholar
  10. 10.
    V.P. Orevkov, A constructive mapping from the square onto itself displacing every constructive point, Soviet Math. Doklady 4(1963), pp. 1253–1256.zbMATHGoogle Scholar
  11. 11.
    N. Shioji, K. Tanaka, Fixed point theory in weak second-order arithmetic, Ann. Pure App. Logic, 47(1990), pp. 167–188.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S.G. Simpson, Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer Verlag, Berlin etc., 1999.Google Scholar
  13. 13.
    W. Veldman, Brouwer’s Fan Theorem as an axiom and as a contrast to Kleene’s alternative, Report No. 0509, Department of Mathematics, Faculty of Science, Radboud University Nijmegen, July 2005.Google Scholar
  14. 14.
    W. Veldman, Brouwer’s Real Thesis on Bars, Philosophia Scientiae, Cahier Spécial 6(2006) 21–39.Google Scholar
  15. 15.
    W. Veldman, Some consequences of Brouwer’s thesis on bars, in: One Hundred years of Intuitionism (1907–2007), The Cerisy Conference, ed. M. van Atten, P. Boldini, M. Bourdeau, G. Heinzmann, Birkhäuser, Basel etc., 2008, pp. 326–340.Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Wim Veldman
    • 1
  1. 1.Institute for Mathematics Astrophysics and Particle Physics Faculty of ScienceRadboud University Nijmegenthe Netherlands

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