Archive for Mathematical Logic

, Volume 42, Issue 6, pp 515–552

Proof-theoretic analysis by iterated reflection

Article

DOI: 10.1007/s00153-002-0158-7

Cite this article as:
Beklemishev, L. Arch. Math. Logic (2003) 42: 515. doi:10.1007/s00153-002-0158-7

Abstract.

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. 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 Π10 and, similarly, for any class Πn0. We provide a more general version of the fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣn, IΠn 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 Σ1-reflection principle for T is Σ2-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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Steklov Mathematical Institute Gubkina 8MoscowRussia