A Calibration of Ineffective Theorems of Analysis in a Hierarchy of Semi-classical Logical Principles

  • Michael Toftdal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


We classify a number of nonconstructive mathematical theorems by means of a hierarchy of logical principles which are not included in intuitionistic logic. The main motivation is the development of the logical hierarchy by Akama Y. et al. and its connection to the so-called limit computable mathematics and proof animations developed by Hayashi S. et al. The results presented give insights in both the scope of limit computable mathematics and its subsystems, and the nature of the theorems of classical mathematics considered.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akama, Y., Berardi, S., Hayashi, S., Kohlenbach, U.: An arithmetical hierachy of the laws of excluded middle and related principles. In: Proc. of the 19th Annual IEEE Symposium on Logic in Computer Science, LICS 2004 (2004) (to appear)Google Scholar
  2. 2.
    Bishop, E.: Foundations of constructive analysis. McGraw-Hill Book Co., New York (1967)MATHGoogle Scholar
  3. 3.
    Bridges, D.S.: A constructive look at functions of bounded variation. Bull. London Math. Soc. 32(3), 316–324 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hayashi, S., Nakata, M.: Towards limit computable mathematics. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 125–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Ishihara, H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36(3), 237–240 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ishihara, H.: Informal Constructive Reverse Mathematics. Research Report Series 229, Centre for Discrete Mathematics and Theoretical Computer Science (2004)Google Scholar
  7. 7.
    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(1), 27–94 (1993)MATHMathSciNetGoogle Scholar
  8. 8.
    Kohlenbach, U.: Relative constructivity. J. Symbolic Logic 63(4), 1218–1238 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kohlenbach, U.: Things that can and things that can’t be done in PRA. Annals of Pure and Applied Logic 102(3), 223–245 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mandelkern, M.: Constructive continuity. Mem. Amer. Math. Soc. 42(277) (1983)Google Scholar
  11. 11.
    Mandelkern, M.: Limited omniscience and the Bolzano-Weierstrass principle. Bull. London Math. Soc. 20(4), 319–320 (1988)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pour-El, M.B., Richards, J.I.: Computability in analysis and physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)MATHGoogle Scholar
  13. 13.
    Richman, F.: Omniscience principles and functions of bounded variation. MLQ Math. Log. Q. 48(1), 111–116 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Shoenfield, J.R.: Recursion theory, Urbana, IL. Lecture Notes in Logic. Association for Symbolic Logic, vol. 1 (2001); Reprint of the 1993 originalGoogle Scholar
  15. 15.
    Simpson, S.G.: Subsystems of second order arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)MATHGoogle Scholar
  16. 16.
    Specker, E.: Der Satz vom Maximum in der rekursiven Analysis. In: Heyting, A. (ed.) Constructivity in mathematics: Proceedings of the colloquium, Amsterdam, 1957. Studies in Logic and the Foundations of Mathematics, pp. 254–265. North-Holland Publishing Co., Amsterdam (1959)Google Scholar
  17. 17.
    Toftdal, M.: Calibration of ineffective theorems of analysis in a constructive context. Master’s thesis, Department of Computer Science, University of Aarhus (May 2004)Google Scholar
  18. 18.
    Troelstra, A.S. (ed.): Metamathematical investigation of intuitionistic arithmetic and analysis. Springer, Berlin (1973)MATHGoogle Scholar
  19. 19.
    Troelstra, A.S., van Dalen, D.: Constructivism in mathematics. Studies in Logic and the Foundations of Mathematics, vol. 121. North-Holland Publishing Co., Amsterdam (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael Toftdal
    • 1
  1. 1.Department of Computer ScienceUniversity of AarhusAarhus NDenmark

Personalised recommendations