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A note on the interpretability logic of finitely axiomatized theories

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Abstract

In [6] Albert Visser shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in finitely axiomatized theories. In this paper we introduce a system called ILP ω that completely axiomatizes the arithmetically valid principles of provability in and interpretability over such theories. To prove the arithmetical completeness of ILP ω we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result.

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References

  1. Dick de Jongh and Frank Veltman. Provability Logics for Relative Interpretability. In: P.P. Petkov (ed.) Mathematical Logic, Proceedings of the 1988 Heyting Conference, Plenum Press, New York, 1990, 31–42.

    Google Scholar 

  2. P. Pudlák. Cuts, Consistency and Interpretations. Journal of Symbolic Logic 50 (1985), 423–441.

    Google Scholar 

  3. Craig Smoryński. Self-Reference and Modal Logic. Springer Verlag, New York, 1985.

    Google Scholar 

  4. Albert Visser. The Provability Logics of Recursively Enumerable Theories Extending Peano Arithmetic at Arbitrary Theories Extending Peano Arithmetic. Journal of Philosophical Logic 13 (1984), 97–113.

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  5. Albert Visser. Preliminary Notes on Interpretability Logic. Logic Group Preprint Series No. 14, Department of Philosophy, University of Utrecht, 1988.

  6. Albert Visser. Interpretability Logic. In: P.P. Petkov (ed.) Mathematical Logic, Proceedings of the 1988 Heyting Conference, Plenum Press, New York, 1990, 175–210.

    Google Scholar 

  7. A.J. Wilkie and J.B. Paris. On the Scheme of Induction for Bounded Arithmetic Formulas. Annals of Pure and Applied Logic 35 (1987), 261–302.

    Google Scholar 

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Research supported by the Netherlands Organization for Scientific Research (NWO).

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de Rijke, M. A note on the interpretability logic of finitely axiomatized theories. Stud Logica 50, 241–250 (1991). https://doi.org/10.1007/BF00370185

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  • DOI: https://doi.org/10.1007/BF00370185

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