24 found
Lev D. Beklemishev [24]Lev Dmitrievich Beklemishev [2]
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  1.  17
    Reflection algebras and conservation results for theories of iterated truth.Lev D. Beklemishev & Fedor N. Pakhomov - 2022 - Annals of Pure and Applied Logic 173 (5):103093.
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  2.  35
    Proof-theoretic analysis by iterated reflection.Lev D. Beklemishev - 2003 - Archive for Mathematical Logic 42 (6):515-552.
    Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity Π1 0 and, similarly, for any class Π n 0 . We provide a more general version of the (...)
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  3.  27
    Provability algebras and proof-theoretic ordinals, I.Lev D. Beklemishev - 2004 - Annals of Pure and Applied Logic 128 (1-3):103-123.
    We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has been (...)
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  4.  22
    Induction rules, reflection principles, and provably recursive functions.Lev D. Beklemishev - 1997 - Annals of Pure and Applied Logic 85 (3):193-242.
    A well-known result states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑n formulas is equivalent to the uniform reflection principle for ∑n + 1 formulas . We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑n formulas is equivalent to ω times iterated ∑n reflection principle. Moreover, for k < ω, k (...)
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  5.  34
    Kripke semantics for provability logic GLP.Lev D. Beklemishev - 2010 - Annals of Pure and Applied Logic 161 (6):756-774.
    A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . First, we isolate a certain subsystem inlMMLBox (...)
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  6.  26
    A proof-theoretic analysis of collection.Lev D. Beklemishev - 1998 - Archive for Mathematical Logic 37 (5-6):275-296.
    By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by Sieg (...)
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  7.  16
    On Provability Logics with Linearly Ordered Modalities.Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten - 2014 - Studia Logica 102 (3):541-566.
    We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...)
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  8.  28
    On the induction schema for decidable predicates.Lev D. Beklemishev - 2003 - Journal of Symbolic Logic 68 (1):17-34.
    We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...)
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  9.  26
    On bimodal logics of provability.Lev D. Beklemishev - 1994 - Annals of Pure and Applied Logic 68 (2):115-159.
    We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories . Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to . Here we study pairs of theories such that the gap between and is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such (...)
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  10.  28
    Bimodal logics for extensions of arithmetical theories.Lev D. Beklemishev - 1996 - Journal of Symbolic Logic 61 (1):91-124.
    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 (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.
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  11.  11
    On the limit existence principles in elementary arithmetic and Σ n 0 -consequences of theories.Lev D. Beklemishev & Albert Visser - 2005 - Annals of Pure and Applied Logic 136 (1-2):56-74.
    We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π1-induction schema. This result, however, cannot be much improved: by adapting a (...)
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  12.  50
    On propositional quantifiers in provability logic.Sergei N. Artemov & Lev D. Beklemishev - 1993 - Notre Dame Journal of Formal Logic 34 (3):401-419.
  13.  13
    [Omnibus Review].Lev D. Beklemishev - 1993 - Journal of Symbolic Logic 58 (2):715-717.
    Reviewed Works:Dick de Jongh, Franco Montagna, Provable Fixed Points.Dick de Jongh, Franco Montagna, Much Shorter Proofs.Alessandra Carbone, Franco Montagna, Rosser Orderings in Bimodal Logics.Alessandra Carbone, Franco Montagna, Much Shorter Proofs: A Bimodal Investigation.
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  14.  25
    On the complexity of arithmetical interpretations of modal formulae.Lev D. Beklemishev - 1993 - Archive for Mathematical Logic 32 (3):229-238.
  15.  8
    Advances in Modal Logic 8.Lev Dmitrievich Beklemishev, Valentin Goranko & Valentin Shehtman (eds.) - 2010 - London, England: College Publications.
    Proc. of the 8th International Conference on Advances in Modal Logic, (AiML'2010).
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  16.  4
    REVIEWS-Godel's theorem: An incomplete guide to its use and abuse.I. Franzen & Lev D. Beklemishev - 2007 - Bulletin of Symbolic Logic 13 (2):241.
  17.  27
    Induction Rules, Reflection Principles, and Provably Recursive Functions.Volker Halbach & Lev D. Beklemishev - 2002 - Bulletin of Symbolic Logic 8 (2):302.
  18.  13
    Dick de Jongh and Franco Montagna. Provable fixed points. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 34 , pp. 229–250. [REVIEW]Lev D. Beklemishev - 1993 - Journal of Symbolic Logic 58 (2):715-717.
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  19.  13
    A many-sorted variant of Japaridze’s polymodal provability logic.Gerald Berger, Lev D. Beklemishev & Hans Tompits - 2018 - Logic Journal of the IGPL 26 (5):505-538.
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  20.  12
    2002 european summer meeting of the association for symbolic logic logic colloquium'02.Lev D. Beklemishev, Stephen Cook, Olivier Lessmann, Simon Thomas, Jeremy Avigad, Arnold Beckmann, Tim Carlson, Robert L. Constable & Kosta Došen - 2003 - Bulletin of Symbolic Logic 9 (1):71.
  21.  13
    Barcelona, Catalonia, Spain July 11–16, 2011.Georges Gonthier, Martin Ziegler, Steve Awodey, George Barmpalias & Lev D. Beklemishev - 2012 - Bulletin of Symbolic Logic 18 (3).
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  22.  9
    Provable Fixed Points.Much Shorter Proofs.Rosser Orderings in Bimodal Logics.Much Shorter Proofs: A Bimodal Investigation. [REVIEW]Lev D. Beklemishev, Dick de Jongh, Franco Montagna & Alessandra Carbone - 1993 - Journal of Symbolic Logic 58 (2):715.
  23.  15
    Provability, complexity, grammars.Lev Dmitrievich Beklemishev - 1999 - Providence, RI: American Mathematical Society. Edited by Mati Reĭnovich Pentus & Nikolai Konstantinovich Vereshchagin.
    (2) Vol., Classification of Propositional Provability Logics LD Beklemishev Introduction Overview. The idea of an axiomatic approach to the study of ...
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  24.  21
    Inexhaustibility: a non-exhaustive treatment and a survey on transfinite progressions. [REVIEW]Lev D. Beklemishev - 2008 - Bulletin of Symbolic Logic 14 (2):258-259.