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

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Automata, Languages and Programming (ICALP 2004)

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

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Abstract

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.

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References

  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. Bishop, E.: Foundations of constructive analysis. McGraw-Hill Book Co., New York (1967)

    MATH  Google Scholar 

  3. Bridges, D.S.: A constructive look at functions of bounded variation. Bull. London Math. Soc. 32(3), 316–324 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  5. Ishihara, H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36(3), 237–240 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ishihara, H.: Informal Constructive Reverse Mathematics. Research Report Series 229, Centre for Discrete Mathematics and Theoretical Computer Science (2004)

    Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  8. Kohlenbach, U.: Relative constructivity. J. Symbolic Logic 63(4), 1218–1238 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mandelkern, M.: Constructive continuity. Mem. Amer. Math. Soc. 42(277) (1983)

    Google Scholar 

  11. Mandelkern, M.: Limited omniscience and the Bolzano-Weierstrass principle. Bull. London Math. Soc. 20(4), 319–320 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pour-El, M.B., Richards, J.I.: Computability in analysis and physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)

    MATH  Google Scholar 

  13. Richman, F.: Omniscience principles and functions of bounded variation. MLQ Math. Log. Q. 48(1), 111–116 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shoenfield, J.R.: Recursion theory, Urbana, IL. Lecture Notes in Logic. Association for Symbolic Logic, vol. 1 (2001); Reprint of the 1993 original

    Google Scholar 

  15. Simpson, S.G.: Subsystems of second order arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)

    MATH  Google Scholar 

  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. 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. Troelstra, A.S. (ed.): Metamathematical investigation of intuitionistic arithmetic and analysis. Springer, Berlin (1973)

    MATH  Google Scholar 

  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)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Aarhus, IT-parken, Aabogade 34, DK-8200, Aarhus N, Denmark

    Michael Toftdal

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  1. Michael Toftdal
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega, 240 Jordi Girona Salgado, 1-3 E-08034, Barcelona

    Josep Díaz

  2. Department of Mathematics and Turku Centre for Computer Science TUCS, University of Turku, 20014, Turku, Finland

    Juhani Karhumäki

  3. Department of Mathematics, University of Turku, Turku, Finland

    Arto Lepistö

  4. Laboratory for Foundations of Computer Science, University of Edinburgh,  

    Donald Sannella

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Toftdal, M. (2004). A Calibration of Ineffective Theorems of Analysis in a Hierarchy of Semi-classical Logical Principles. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_98

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  • DOI: https://doi.org/10.1007/978-3-540-27836-8_98

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22849-3

  • Online ISBN: 978-3-540-27836-8

  • eBook Packages: Springer Book Archive

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