Skip to main content
×
×
Home

Bimodal logics for extensions of arithmetical theories

  • Lev D. Beklemishev (a1)
Abstract

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ0 + EXP, PRA); (PRA, IΣn); (IΣm, IΣn) for 1 ≤ m < n; (PA, ACA0); (ZFC, ZFC + CH); (ZFC, ZFC + ¬CH) etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

Copyright
References
Hide All
[1]Artemov, S. N., Arithmetically complete modal theories, Semiotika i Informatika, vol. 14 (1980), pp. 115133, in Russian. English translation in: American Mathematical Society Translations, Series 2, 135: 39–54, 1987.
[2]Beklemishev, L. D., On the classification of propositional provability logics, Izvestiya Akademii Nauk SSSR, ser. mat., vol. 53 (1989), no. 5, pp. 915943, in Russian. English translation in Mathematics of the USSR-Izvestiya, vol. 35 (1990), pp. 247–275.
[3]Beklemishev, L. D., Provability logics for natural Turing progressions of arithmetical theories, Studia Logica, vol. L (1991), no. 1, pp. 107128.
[4]Beklemishev, L. D., Independent numerations of theories and recursive progressions, Sibirskii Matematicheskii Zhurnal, vol. 33 (1992), pp. 2246, in Russian. English translation in Siberian Mathematical Journal, vol. 33 (1992), pp. 760–783.
[5]Beklemishev, L. D., On bimodal logics of provability, Annals of Pure and Applied Logic, vol. 68 (1994), no. 2, pp. 115159.
[6]Berarducci, A. and Verbrugge, R., On the provability logic of bounded arithmetic, Annals of Pure and Applied Logic, vol. 61 (1993), pp. 7593.
[7]Boolos, G., The analytical completeness of Dzhaparidze's polymodal logics, Annals of Pure and Applied Logic, vol. 61 (1993), pp. 95111.
[8]Boolos, G., The logic of provability, Cambridge University Press, Cambridge, 1993.
[9]Carlson, T., Modal logics with several operators and provability interpretations, Israel Journal of Mathematics, vol. 54 (1986), pp. 1424.
[10]Dzhaparidze, G. K., The modal logical means of investigation of provability, Ph.D. thesis, Moscow State University, 1986.
[11]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae vol. 49 (1960), pp. 3592.
[12]Guaspari, D., Partially conservative sentences and interpretability, Transactions of the American Mathematical Society, vol. 254 (1979), pp. 4768.
[13]Guaspari, D. and Solovay, R., Rosser sentences, Annals of Mathematical Logic, vol. 16 (1979), pp. 8199.
[14]Hájek, P. and Pudlák, P., Metamathematics of first order arithmetic, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
[15]Ignatiev, K. N., On strong provability predicates and the associated modal logics, this Journal, vol. 58 (1993), pp. 249290.
[16]Ignjatovic, A. D., Fragments of first and second order arithmetic and length of proof, Ph.D. thesis, University of California at Berkeley, 1990.
[17]Kreisel, G. and Lévy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97142.
[18]Leivant, D., The optimality of induction as an axiomatization of arithmetic, this Journal, vol. 48 (1983), pp. 182184.
[19]Lindström, P., On partially conservative sentences and interpretability, Proceedings of the American Mathematical Society, vol. 91 (1984), no. 3, pp. 436443,
[20]Montagna, F., Provability in finite subtheories of PA, this Journal, vol. 52 (1987), no. 2, pp. 494511.
[21]Parsons, C., On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory (Kino, , Myhill, , and Vessley, , editors), North Holland, Amsterdam, 1970, pp. 459473.
[22]Pozsgay, L. J., Gödel's second theorem for elementary arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 6780.
[23]Rose, H. E., Subrecursion: Functions and hierarchies, Clarendon Press, Oxford, 1984.
[24]Schmerl, U. R., A fine structure generated by reflection formulas over primitive recursive arithmetic, Logic colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North Holland, Amsterdam, 1979, pp. 335350.
[25]Schwichtenberg, H., Some applications of cut-elimination, Handbook of mathematical logic (Barwise, J., editor), North Holland, Amsterdam, 1977, pp. 867896.
[26]Shavrukov, V. Yu., On Rosser's provability predicate, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 37 (1991), pp. 317330.
[27]Shavrukov, V. Yu., A smart child of Peano's, Notre Dame Journal of Formal Logic, vol. 35 (1994), no. 2, pp. 161185.
[28]Shoenfield, J. R., Mathematical logic, Addison-Wesley Publishing Company, 1967.
[29]Sieg, W., Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 3371.
[30]Smoryński, C., The incompleteness theorems, Handbook of mathematical logic (Barwise, J., editor), North Holland, Amsterdam, 1977, pp. 821865.
[31]Smoryński, C., Self-reference and modal logic, Springer-Verlag, Berlin, Heidelberg, New York, 1985.
[32]Solovay, R. M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 28 (1976), pp. 3371.
[33]Visser, A., Peano's smart children. A provability logical study of systems with built-in consistency, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 161196.
[34]Visser, A., Interpretability logic, Mathematical logic (Petkov, P. P., editor), Plenum Press, New York, 1990, pp. 175208.
[35]Petkov, P. P., A course in bimodal provability logic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 109142.
[36]Wilkie, A. and Paris, J., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261302.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×