Abstract
A nonempty sequence 〈T1,...,Tn〉 of theories is tolerant, if there are consistent theories T +1 ,..., T +n such that for each 1 ≤i ≤n, T +i is an extension of Ti in the same language and, if i ≤n, T +i interprets T +i+1 . We consider a propositional language with the modality ◊, the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOLω. It is shown that TOL (resp. TOLω) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if ◊(A1,..., An) is understood as (a formalization of) “〈 PA+A1, ..., PA+An〉 is tolerant”.
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References
A. Berarducci, The interpretability logic of Peano arithmetic, Entailment: The Journal of Symbolic Logic, 55 (1990), No.3, pp. 1059–1089.
G. Boolos, The Unprovability of Consistency, Cambridge Universitety Press, Cambridge, 1979.
D. De Jongh, and F. Veltman, Provability logics for relative interpretability. To apper in the Proceedings of Heyting'88 Conference, Chaika, Bulgaria, 1988.
G. Dzhaparidze, Provability logic with modalities for arithmetical complexities, Bulletin of the Academy of Sciences of the Georgian SSR 138 (1990), No. 3. pp. 481–484.
S. Peferman, Arithmetization of Metamathematics in a general setting, Fundamenta Mathematicae, 49 (1990), pp. 35–92.
D. Guaspari, Partially conservative extentions of arithmetic, Transactions of the Amer. Math. Soc. 254 (1979), pp. 47–68.
P. Hajek On interpretability in set theories I, II Comm. Math. Univ. Carolinae, 12 (1971), pp. 73–79 and 13 (1972), pp. 445–455.
K. Ignatiev, Logic of ∑1 -interpolability over Peano arithmetic, (in Russian), Manuscript, Moscow, September 1990.
S. Orey, Relative interpretations, Zeitschrift für Math. Logik und Grundlagen der Mathematik, 7 (1961), pp. 146–153.
V. Shavrukow, Logic of relative interpretability over Peano arithmetic, Preprint No. 5, Steklov Mathematical Institute, Academy of Sciences of the USSR, Moscow, December 1988.
R. M. Solovay, Provability interpretations of logic, Israel Journal of Mathematics, 25 (1976), pp. 287–304.
A. Tarski, in colab with A. Mostowski and R. M. Robinson, Undecidable theories, Amsterdam, 1953.
A. Visser Preliminary notes on interpretability logic, Logic group preprint series, No. 29, Department of Philosophy, University of Utrecht, Utrecht, 1988.
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Dzhaparidze, G. The logic of linear tolerance. Stud Logica 51, 249–277 (1992). https://doi.org/10.1007/BF00370116
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DOI: https://doi.org/10.1007/BF00370116