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Proof Theoretic Analysis by Iterated Reflection

  • L. D. Beklemishev

Abstract

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. Moreover, they provide a uniform definition of a proof-theoretic ordinal for any arithmetical complexity \(\Pi _{n}^{0}\). 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 \(\Pi _{1}^{0}\). We provide a more general version of the fine structure relationships for iterated reflection principles (due to Ulf Schmerl). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including \(I\Sigma _{n}\), \(I\Sigma _{n}^{-}\), \(I\Pi _{n}^{-}\) and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform \(\Sigma _{1}\)-reflection principle for T is \(\Sigma _{2}\)-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem.

Keywords

Ordinal analysis Reflection principles Turing progressions Partial conservativity Parameter-free induction 

Notes

Acknowledgements

The bulk of this paper was written during my stay in 1998–1999 as Alexander von Humboldt fellow at the Institute for Mathematical Logic of the University of Münster. Discussions with and encouragements from W. Pohlers, A. Weiermann, W. Burr, and M. Möllerfeld have very much influenced both the ideological and the technical sides of this work. I would also like to express my cordial thanks to W. and R. Pohlers, J. and D. Diller, H. Brunstering, M. Pfeifer, W. Burr, A. Beckmann, B. Blankertz, I. Lepper, and K. Wehmeier for their friendly support during my stay in Münster.

Supported by Alexander von Humboldt Foundation, INTAS grant 96-753, and RFBR grants 98-01-00282 and 15-01-09218.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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