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Lifschitz' realizability

  • Jaap van Oosten (a1)
Abstract

V. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a “q-variant” we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe an analogous development for elementary analysis, with partial continuous application replacing partial recursive application.

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References
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Dragalin, A. G. [1979], Mathematical intuitionisnu introduction to proof theory, “Nauka”, Moscow; English translation, American Mathematical Society, Providence, Rhode Island, 1988.
Hyland, J. M. E. [1982], The effective topos, The L.E.J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North-Holland, Amsterdam, pp. 165216.
Kleene, S. C. [1969], Formalized recursive functionals and formalized realizability, Memoirs of the American Mathematical Society, no. 89 (1969).
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Lifschitz, V. [1979], CT0is stronger than CT0!, Proceedings of the American Mathematical Society, vol. 73, pp. 101106.
Troelstra, A. S. (editor) [1973], Metamathematical investigation of intuitionistic arithmetic and analysis, Lectures Notes in Mathematics, vol. 344, Springer-Verlag, Berlin.
Troelstra, A. S. and van Dalen, D. [1988], Constructivism in mathematics, North-Holland, Amsterdam.
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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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