A simplified proof of arithmetical completeness theorem for provability logic GLP

  • L. D. Beklemishev


We present a simplified proof of Japaridze’s arithmetical completeness theorem for the well-known polymodal provability logic GLP. The simplification is achieved by employing a fragment J of GLP that enjoys a more convenient Kripke-style semantics than the logic considered in the papers by Ignatiev and Boolos. In particular, this allows us to simplify the arithmetical fixed point construction and to bring it closer to the standard construction due to Solovay.


STEKLOV Institute Kripke Model Axiom Schema Modal Formula Kripke Frame 
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  1. 1.
    L. D. Beklemishev, “Reflection Principles and Provability Algebras in Formal Arithmetic,” Usp. Mat. Nauk 60(2), 3–78 (2005) [Russ. Math. Surv. 60, 197–268 (2005)].MathSciNetGoogle Scholar
  2. 2.
    L. D. Beklemishev, “Kripke Semantics for Provability Logic GLP,” Ann. Pure Appl. Logic 161, 756–774 (2010); Logic Group Preprint Ser. 260 (Univ. Utrecht, Oct. 2007), MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    G. Boolos, The Logic of Provability (Cambridge Univ. Press, Cambridge, 1993).zbMATHGoogle Scholar
  4. 4.
    K. N. Ignatiev, “On Strong Provability Predicates and the Associated Modal Logics,” J. Symb. Log. 58, 249–290 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. K. Japaridze, “The Modal Logical Means of Investigation of Provability,” Candidate (Philos.) Dissertation (Moscow State Univ., Moscow, 1986).Google Scholar
  6. 6.
    C. Smoryński, “The Incompleteness Theorems,” in Handbook of Mathematical Logic, Ed. by J. Barwise (North Holland, Amsterdam, 1977), pp. 821–865.CrossRefGoogle Scholar
  7. 7.
    R. M. Solovay, “Provability Interpretations of Modal Logic,” Isr. J. Math. 25, 287–304 (1976).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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