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Bounded modified realizability

  • Fernando Ferreira (a1) and Ana Nunes (a2) (a3)

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov's principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König's lemma. We discuss some applications, as well as some previous results in the literature.

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[1]Akama, Y., Berardi, S., Hayashi, S., and Kohlenbach, U., An arithmetical hierarchy of the law of excluded middle and related principles, Proceedings of the 19th annual IEEE symposium on logic and computer science, vol. 117, IEEE Press, 2004, pp. 192201.
[2]Avigad, J. and Feferman, S., Gödel's functional (“Dialectica”) interpretation, Handbook of proof theory (Buss, S. R., editor), Studies in Logic and the Foundations of Mathematics, vol. 137, North Holland, Amsterdam, 1998, pp. 337405.
[3]Bezem, M., Strongly majorizable functional of finite type: a model for bar recursion containing discontinuous Junctionals. this Journal, vol. 50 (1985), pp. 652660.
[4]Bridges, D. S. and Richman, F., Varieties of constructive mathematics, London Mathematical Society Lecture Notes Series, vol. 97. Cambridge University Press, 1987.
[5]Ferreira, F. and Oliva, P., Bounded functional interpretation, Annals of Pure and Applied Logic, vol. 135 (2005), pp. 73112.
[6]Gödel, K., Über eine bisher noch nicht benützte Erweiterung desfiniten Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.
[7]Howard, W. A., Hereditarily majorizable functional of finite type, Metamathematical investigation of intuitionistic Arithmetic and Analysis (Troelstra, A. S., editor), Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973, pp. 454461.
[8]Kleene, S. C., On the interpretation of intuitionistic number theory, this Journal, vol. 10 (1945), pp. 109124.
[9]Kohlenbach, U., Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals, Archive for Mathematical Logic, vol. 36 (1996), pp. 3171.
[10]Kohlenbach, U., Relative constructivity, this Journal, vol. 63 (1998), pp. 12181238.
[11]Kohlenbach, U., On uniform weak König's lemma, Annals of Pure Applied Logic, vol. 114 (2002), pp. 103116.
[12]Kohlenbach, U. and Oliva, P., Proof mining: a systematic way of analysing proofs in mathematics, Proceedings of the Steklov Institute of Mathematics, vol. 242, 2003, pp. 136164.
[13]Kreisel, G., On weak completeness of intuitionistic predicate logic, this Journal, vol. 27 (1962), pp. 139158.
[14]Troelstra, A. S. (editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.
[15]Troelstra, A. S., Introductory note to 1958 and 1972, Kurt Gödel collected works (Feferman, Solomonet al., editors), vol. II, Oxford University Press, New York, 1990, pp. 217241.
[16]Troelstra, A. S., Realizability, Handbook of proof theory (Buss, S. R., editor), vol. 137, North Holland, Amsterdam, 1998, pp. 408473.
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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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