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Proof-theoretic analysis by iterated reflection

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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 Π1 0 and, similarly, for any class Π n 0. 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 IΣ n , IΣ n −, IΠ 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 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.

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  1. Steklov Mathematical Institute Gubkina 8, 117966, Moscow, Russia

    L.D. Beklemishev

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Correspondence to L.D. Beklemishev.

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Supported by Alexander von Humboldt Foundation, INTAS grant 96-753, and RFBR grant 8-01-00282.

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Beklemishev, L. Proof-theoretic analysis by iterated reflection. Arch. Math. Logic 42, 515–552 (2003). https://doi.org/10.1007/s00153-002-0158-7

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