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We study reflection principles in fragments of Peano arithmetic and their applications to the questions of comparison and classification of arithmetical theories. Bibliography: 95 items.

- Lev D. Beklemishev
- Ann. Pure Appl. Logic
- 2004

- Lev D. Beklemishev, Yuri Gurevich
- J. Log. Comput.
- 2014

Gurevich and Neeman introduced Distributed Knowledge Authorization Language (DKAL). The world of DKAL consists of communicating principals computing their own knowledge in their own states. DKAL is based on a new logic of information, the so-called infon logic, and its efficient subsystem called primal logic. In this paper we simplify Kripkean semantics of… (More)

- Lev D. Beklemishev
- Ann. Pure Appl. Logic
- 2010

A well-known polymodal provability logic GLP is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic [9, 5, 8]. This system plays an important role in some recent applications of provability algebras in proof theory [2, 3]. However, an obstacle in the study of GLP is… (More)

We study an extension of Japaridze’s polymodal logic GLP with transfinitely many modalities and develop a provability-algebraic ordinal notation system up to the ordinal Γ0. In the papers [1, 2] a new algebraic approach to the traditional prooftheoretic ordinal analysis was presented based on the concept of graded provability algebra. The graded provability… (More)

- Lev D. Beklemishev
- Arch. Math. Log.
- 2003

Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omega-rule. We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques. In some cases the techniques of iterated reflection… (More)

- Lev D. Beklemishev
- Theor. Comput. Sci.
- 1999

We study the classes of computable functions that can be proved to be total by means of parameter free C, and 4 induction schemata, ZC; and ZlI;, over Kalmar elementary arithmetic. We give a positive answer to a question, whether the provably total computable functions of Zq are exactly the primitive recursive ones, and show that the class of such functions… (More)

- Lev D. Beklemishev
- Ann. Pure Appl. Logic
- 1997

A well known result of D Leivant states that over basic Kalmar ele mentary arithmetic EA the induction schema for n formulas is equivalent to the uniform re ection principle for n formulas We show that frag ments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of re ection principles as well Thus the… (More)

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… (More)

- Lev D. Beklemishev
- Kurt Gödel Colloquium
- 1997

We give a precise characterization of parameter free n and n induction schemata I n and I n in terms of re ection principles This allows us to show that I n is conservative over I n w r t boolean combinations of n sentences for n In particular we give a positive answer to a question by R Kaye whether the provably recursive functions of I are exactly the… (More)