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  1.  22
    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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  2.  12
    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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  3.  20
    Topological Completeness of the Provability Logic GLP.Lev Beklemishev & David Gabelaia - 2013 - Annals of Pure and Applied Logic 164 (12):1201-1223.
    Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.
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  4.  17
    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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  5.  12
    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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  6.  28
    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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  7.  22
    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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  8.  11
    Positive Provability Logic for Uniform Reflection Principles.Lev Beklemishev - 2014 - Annals of Pure and Applied Logic 165 (1):82-105.
    We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n<ω corresponds to its restriction to arithmetical Πn+1-formulas. This (...)
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  9.  11
    Iterated Local Reflection Versus Iterated Consistency.Lev Beklemishev - 1995 - Annals of Pure and Applied Logic 75 (1-2):25-48.
    For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...)
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  10.  18
    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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  11.  8
    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.  46
    Notes on Local Reflection Principles.Lev Beklemishev - 1997 - Theoria 63 (3):139-146.
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  13.  44
    Provability Logics for Natural Turing Progressions of Arithmetical Theories.L. D. Beklemishev - 1991 - Studia Logica 50 (1):107 - 128.
    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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  14.  15
    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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  15.  6
    [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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  16.  44
    On Propositional Quantifiers in Provability Logic.Sergei N. Artemov & Lev D. Beklemishev - 1993 - Notre Dame Journal of Formal Logic 34 (3):401-419.
  17.  15
    Axiomatization of Provable N-Provability.Evgeny Kolmakov & Lev Beklemishev - 2019 - Journal of Symbolic Logic 84 (2):849-869.
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  18.  16
    Wolfgang Burr. Fragments of Heyting Arithmetic. The Journal of Symbolic Logic, Vol. 65 , Pp. 1223–1240. [REVIEW]Lev Beklemishev - 2002 - Bulletin of Symbolic Logic 8 (4):533-534.
  19.  20
    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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  20.  22
    On the Complexity of Arithmetical Interpretations of Modal Formulae.Lev D. Beklemishev - 1993 - Archive for Mathematical Logic 32 (3):229-238.
  21.  11
    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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  22.  25
    Franco Montagna’s Work on Provability Logic and Many-Valued Logic.Lev Beklemishev & Tommaso Flaminio - 2016 - Studia Logica 104 (1):1-46.
    Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.
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  23.  5
    Smullyan Raymond M.. Diagonalization and Self-Reference. Oxford Logic Guides, No. 27. Clarendon Press, Oxford University Press, Oxford and New York1994, Xv + 396 Pp. [REVIEW]Lev Beklemishev - 1996 - Journal of Symbolic Logic 61 (3):1052-1055.
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  24.  20
    Induction Rules, Reflection Principles, and Provably Recursive Functions.Volker Halbach & Lev D. Beklemishev - 2002 - Bulletin of Symbolic Logic 8 (2):302.
  25.  24
    Review: Raymond M. Smullyan, Diagonalization and Self-Reference. [REVIEW]Lev Beklemishev - 1996 - Journal of Symbolic Logic 61 (3):1052-1055.
  26.  13
    Preface.Uri Abraham, Lev Beklemishev, Paola D'Aquino & Marcus Tressl - 2016 - Annals of Pure and Applied Logic 167 (10):865-867.
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  27.  6
    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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  28.  20
    Foreword.Lev Beklemishev, Guram Bezhanishvili, Daniele Mundici & Yde Venema - 2012 - Studia Logica 100 (1-2):1-7.
  29.  9
    Annals of Pure and Applied Logic.Lev Beklemishev - 2001 - Bulletin of Symbolic Logic 7 (1):75-77.
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  30.  10
    Carnegie Mellon University, Pittsburgh, PA May 19–23, 2004.John Baldwin, Lev Beklemishev, Michael Hallett, Valentina Harizanov, Steve Jackson, Kenneth Kunen, Angus J. MacIntyre, Penelope Maddy, Joe Miller & Michael Rathjen - 2005 - Bulletin of Symbolic Logic 11 (1).
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  31.  10
    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.
  32.  10
    18th Workshop on Logic, Language, Information and Computation (Wollic 2011).Lev Beklemishev, Ruy de Queiroz & Andre Scedrov - 2012 - Bulletin of Symbolic Logic 18 (1):152-153.
  33.  10
    Vassar College, 124 Raymond Avenue, Poughkeepsie, Ny 12604, Usa. In a Review, a Reference “Jsl Xliii 148,” for Example, Refers Either to the Publication Reviewed on Page 148 of Volume 43 of the Journal, or to the Review Itself (Which Contains Full Bibliographical Information for the Reviewed Publication). Analogously, a Reference “Bsl VII 376” Refers to the Review Beginning on Page 376 in Volume 7 of This Bulletin, Or. [REVIEW]John Baldwin, Lev Beklemishev, Anuj Dawar, Mirna Dzamonja, David Evans, Erich Grädel, Denis Hirschfeldt, Hannes Leitgeb, Roger Maddux & Grigori Mints - 2008 - Bulletin of Symbolic Logic 14 (1).
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  34.  10
    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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  35.  6
    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.
  36.  8
    Review: Ulrich Kohlenbach, On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness. [REVIEW]Lev Beklemishev - 2001 - Bulletin of Symbolic Logic 7 (1):75-77.
  37.  6
    Review: Per Lindstrom, Aspects of Incompleteness. [REVIEW]Lev Beklemishev - 1998 - Journal of Symbolic Logic 63 (4):1606-1608.
  38.  12
    Leo Corry, David Hilbert and the Axiomatization of Physics (1998–1918), Springer, Netherlands (2004) ISBN 1-4020-2777-X (513 Pp., Euro 160, US$ 179, £111, Hardcover). [REVIEW]L. BekLemishev - 2006 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 37 (2):388-390.
  39.  3
    Lindström Per. Aspects of Incompleteness. Lecture Notes in Logic, No. 10. Springer, Berlin, Heidelberg, New York, Etc., 1997, X + 133 Pp. [REVIEW]Lev Beklemishev - 1998 - Journal of Symbolic Logic 63 (4):1606-1608.
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  40.  2
    Advances in Modal Logic 8.Lev Beklemishev, Valentin Goranko & Valentin B. Shehtman (eds.) - 2010 - College Publications.
    Proc. of the 8th International Conference on Advances in Modal Logic, (AiML'2010).
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  41. Leloup, G., Rings of Monoids Elementarily Equivalent to Polynomial Rings Miller, C., Expansions of the Real Field with Power Functions Ozawa, M., Forcing in Nonstandard Analysis Rathjen, M., Proof Theory of Reflection. [REVIEW]L. D. Beklemishev, O. V. Belegradek, K. J. Davey & J. L. Krivine - 1994 - Annals of Pure and Applied Logic 68:343.
  42.  11
    Provability, Complexity, Grammars.Lev Dmitrievich Beklemishev - 1999 - American Mathematical Society.
    (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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  43. 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.