On bimodal logics of provability

AbstractWe 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 (PA, ZF). 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 pairs (,) that is axiomatized over by ∏1-sentences only, and, for each n ⩾ 1, proves the n-times iterated consistency of . A complete axiomatization, along with the appropriate Kripke semantics and decision procedures, is found for the two principal cases: finitely axiomatizable extensions of this sort, like e.g. (PA, PA + Con(ZF)), (I∑1, I∑1 + Con(I∑2)), etc., and reflexive extensions, like (PRA, PRA + \s{Con(I∑n)¦n ⩾ 1\s}), etc. We show that the first logic, ICP, is the minimal and the second one, RP, is the maximal within the class of the provability logics for such pairs of theories. We also show that there are some provability logics lying strictly between these two. As an application of the results of this paper, in the last section the polymodal provability logics for natural recursive progressions of theories based on iteration of consistency are characterized. We construct a system of ordinal notation , which gives exactly one notation to each constructive ordinal, such that the logic corresponding to any progression along coincides with that along natural Kalmar elementary well-orderings

Parameter free induction and provably total computable functions

AbstractWe study the classes of computable functions that can be proved to be total by means of parameter free Σn and Πn, induction schemata, IΣn− and IΠn−, over Kalmar elementary arithmetic. We give a positive answer to a question, whether the provably total computable functions of IΠ2− are exactly the primitive recursive ones, and show that the class of such functions for IΣ1 + IΠ2− coincides with the class of doubly recursive functions of Peter. We also characterize provably total computable functions of theories of the form IΠn + 1− and IΣn + IΠn + 1− for all n ⩾ 1, in terms of the fast growing hierarchy.These results are based on a precise characterization of IΣn− and IΠn− in terms of reflection principles and conservation results for local reflection principles obtained by techniques of modal provability logic. Using similar ideas we show that IΠn + 1− is conservative over IΣn− w.r.t. boolean combinations of Σn + 1 sentences, for n ⩾ 1, and obtain a number of results on the strength of bounded number of instances of parameter free induction schemata and complexity of their axiomatization

Kripke semantics for provability logic GLP

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 that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for GLP. First, we isolate a certain subsystem J of GLP that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for GLP are defined as the limits of chains of finite expansions of models for J. The techniques involves unions of n-elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic. This paper is devoted to a modal-logical study of polymodal provability logic GLP introduced by Giorgi Japaridze [9, 10] as early as in 1986. This logic describes in the style of provability logic all the universally valid schemata for the reflection principles of restricted logical complexity in arithmetic. Recently, important applications of GLP have been found in proof theory and ordinal analysis of arithmetic, which stimulated further interest towards GLP (see ref. [2] and ref. [3] for a more recent survey). The modal-logical study of GLP was initiated by Konstantin Ignatiev [7, 8] who simplified Japaridze’s arithmetical completeness theorem and established Craig’s interpolation and fixed-point properties for this logic. He also gave a normal form theorem and a universal Kripke model for the close

On the Induction Schema for Decidable Predicates

. We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, I1 . We show that I1 is independent from the set of all true arithmetical 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 establish some conservation and independence results for parameter free and inference rule forms of 1-induction. An open problem formulated by J. Paris (see [4, 5]) is whether I1 proves the corresponding least element principle for decidable predicates, L1 (or, equivalently, the 1 -collection principle B1 ). We reduce this question to a purely computation-theoretic one. 1 Introduction and motivation The schema of induction for decidable predicates I 1 is considered to be rather exotic. Indeed, the stronger schema of induction for r.e. predicates I 1 appears more naturally in the formalization of va..