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The fan theorem and unique existence of maxima

  • Josef Berger (a1), Douglas Bridges (a2) and Peter Schuster (a3)

The existence and uniqueness of a maximum point for a continuous real–valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.

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[1] P. Aczel and M. Rathjen , Notes on constructive set theory. Technical Report 40. Institut Mittag–Leffler. Royal Swedish Academy of Sciences. 2001.

[2] E. Bishop and D. Bridges , Constructive analysis. Grundlehren der Matheraatischen Wissenschaften. vol. 279. Springer–Verlag, Heidelberg, 1985.

[3] D. Bridges , A constructive proximinality property of finite–dimensional linear spaces. Rocky Mountain Journal of Mathematics, vol. 11 (1981), no. 4, pp. 491497.

[4] D. Bridges , Recent progress in constructive approximation theory, The L.E.J. Brouwer Centenary Symposium ( A.S. Troelstra and D. van Dalen , editors). North–Holland, Amsterdam, 1982, pp. 4150.

[7] D. Bridges and F. Richman , Varieties of constructive mathematics, London Mathematical Society Lecture Notes, vol. 97, Cambridge University Press, 1987.

[10] K-I Ko . Complexity theory of real functions, Birkhäuser, Boston–Basel–Berlin. 1991.

[11] U. Kohlenbach , Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Annals of Pure and Applied Logic, vol. 64 (1993), pp. 2794.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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