Abstract
Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved.
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References
T. Carlson, Modal logics with several operators and provability interpretations, Israel Journal of Mathematics 54 (1986), pp. 14–24.
S. Feferman, Transfinite recursive progressions of axiomatic theories, Journal of Symbolic Logic 27 (1962), No. 3, pp. 259–316.
G. Kreisel, Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt, Jahresbericht der Deutschen Mathematiker-Vereinigung 78 (1977), Heft 4, pp. 111–223.
F. Montagna, Provability in finite subtheories of PA, Journal of Symbolic Logic 52 (1987), No. 2, pp. 494–511.
K. Schütte, Beweistheorie, Springer, 1960.
C. Smoryński, Modal Logic and Selfreference, Springer, 1985.
C. Smoryński, Quantified modal logic and selfreference, Notre Dame Journal of Formal Logic 28 (1987), pp. 356–370.
R. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics 25 (1976), pp. 287–304.
A. Visser, A Course in Bimodal Provability Logic, Logic Group Preprint Series 20 (1987), University of Utrecht.
С. Н, Артëмов, Приложения модальной логики в теории доказательств, In Неклассические логики и их применения, Вопросы кибернетики, М. Наука, 1982, pp. 3–20.
Л. Д. Беклемишев, О классификации пропозициональных логик доказуемости, Известия АН СССР, cep. mat. 53 (1989), pp. 915–943.
С. В. Горячев, Об интерпретируемости некоторых расширений арифметики, Математические заметки 40 (1986), pp. 561–572.
U. Schmerl, A fine structure generated by reflection formulas, In Logic Colloquium '78, M. Boffa, D. van Dalen, K. McAloon (eds.), North-Holland, 1979, pp. 335–350.
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Beklemishev, L.D. Provability logics for natural turing progressions of arithmetical theories. Stud Logica 50, 107–128 (1991). https://doi.org/10.1007/BF00370390
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DOI: https://doi.org/10.1007/BF00370390