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AXIOMATIZATION OF PROVABLE n-PROVABILITY

Published online by Cambridge University Press:  08 February 2019

EVGENY KOLMAKOV  and
LEV BEKLEMISHEV
EVGENY KOLMAKOV
Affiliation:
STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES 8 GUBKINA STREET, MOSCOW 119991, RUSSIA and FACULTY OF COMPUTER SCIENCE NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS 3 KOCHNOVSKY PROEZD, MOSCOW 125319, RUSSIAE-mail: kolmakov-ea@yandex.ru
LEV BEKLEMISHEV
Affiliation:
STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES 8 GUBKINA STREET, MOSCOW 119991, RUSSIA and FACULTY OF MATHEMATICS NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS 6 USACHEVA STREET, MOSCOW 119048, RUSSIAE-mail: bekl@mi-ras.ru
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Abstract

A formula φ is called n-provable in a formal arithmetical theory S if φ is provable in S together with all true arithmetical ${{\rm{\Pi }}_n}$-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 ${\varepsilon _0}$ 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.

Keywords

03F2503F3003F4003F45strong provability predicatereflection principlePeano arithmeticTuring progressionn-provability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 849 - 869
Copyright
Copyright © The Association for Symbolic Logic 2019 

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