Reflection principles and provability algebras in formal arithmetic

Beklemishev, Lev D.

doi:10.1070/rm2005v060n02abeh000823

Cited by 85 publications

(130 citation statements)

References 77 publications

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“…Reflection calculus RC is much simpler than its modal companion GLP yet expressive enough for its main proof-theoretic applications. It has been outlined in [9] that RC allows to define a natural system of ordinal notations up to ε 0 and serves as a convenient basis for a proof-theoretic analysis of Peano Arithmetic in the style of [6,7]. This includes a consistency proof for Peano arithmetic based on transfinite induction up to ε 0 , a characterization of its Π 0 n -consequences in terms of iterated reflection principles, a slowly terminating term rewriting system [2] and a combinatorial independence result [8].…”

Section: Introductionmentioning

confidence: 99%

Reflection calculus and conservativity spectra

Beklemishev¹

2018

Russ. Math. Surv.

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Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus RC consists of implications between formulas built up from propositional variables and constant 'true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles.We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity Π 0 n , for all n > 0. We note that such operators, in a strong sense, cannot be represented in the full language of modal logic.We formulate a formal system RC ∇ extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal < ε0. Secondly, we provide normal forms for its variable-free fragment. Thereby, this fragment is shown to be algorithmically decidable and complete w.r.t. its natural arithmetical semantics.In the last part of the paper we characterize in several natural ways the Lindenbaum-Tarski algebra of the variable-free fragment of RC ∇ and its dual Kripke structure. Most importantly, the elements of this algebra correspond to the sequences of proof-theoretic Π 0 n+1 -ordinals of bounded fragments of Peano arithmetic called conservativity spectra, as well as to the points of the well-known Ignatiev Kripke model.

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Section: Introductionmentioning

confidence: 99%

Reflection calculus and conservativity spectra

Beklemishev¹

2018

Russ. Math. Surv.

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“…It was used to characterize the fragments of arithmetic defined by parameter free induction [2] and to study their properties such as complexity of axiomatization, the classes of provably total computable functions, etc. The notion of n-provability and the modal logic GLP also played a prominent role in the approach to the ordinal analysis of systems of arithmetic based on provability algebras [4,5].…”

Section: Introductionmentioning

confidence: 99%

Axiomatizing Provable n-Provability

Kolmakov

Beklemishev

2018

Dokl. Math.

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A formula ϕ is called n-provable in a formal arithmetical theory S if ϕ is provable in S together with all true arithmetical Πn-sentences taken as additional axioms. While in general the set of all n-provable formulas, for a fixed n > 0, is not recursively enumerable, the set of formulas ϕ whose n-provability is provable in a given r.e. metatheory T is r.e. This set is deductively closed and will be, in general, an extension of S. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic PA can be axiomatized by ε0 times iterated local reflection schema over PA. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of n-provability of a sentence can be much shorter than its proof from iterated reflection principles.

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“…The latter equivalent coincides with the lemma in Leivant [7] modulo some ambiguity as to what Leivant's base theory Z 0 exactly is. The verification of EA-provability is best carried out along the proof of TeopeMa 7 in [1]. We note that we have the following as an immediate consequence.…”

Section: Theorem 32mentioning

confidence: 99%

Uniform Density in Lindenbaum Algebras

Shavrukov¹,

Visser²

2014

Notre Dame J. Formal Logic

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In this paper we prove that the preordering . of provable implication over any recursively enumerable theory T containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function F for .. A recursive function F is a density function if it computes, for A and B with A ' B, an element C such that A ' C ' B. The function is extensional if it preserves T -provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering . restricted to † n -sentences is uniformly dense. In the last section we provide historical notes and background material.

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Reflection principles and provability algebras in formal arithmetic

Abstract: 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.

Cited by 85 publications

References 77 publications

Reflection calculus and conservativity spectra

Reflection calculus and conservativity spectra

Axiomatizing Provable n-Provability

Uniform Density in Lindenbaum Algebras

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