Archive for Mathematical Logic

, Volume 37, Issue 5–6, pp 275–296 | Cite as

A proof-theoretic analysis of collection

  • Lev D. Beklemishev


By a result of Paris and Friedman, the collection axiom schema for \(\Sigma_{n+1}\) formulas, \(B\Sigma_{n+1}\), is \(\Pi_{n+2}\) conservative over \(I\Sigma_n\). We give a new proof-theoretic proof of this theorem, which is based on a reduction of \(B\Sigma_n\) to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for \(B\Sigma_n\) and to answer some technical questions left open by Sieg [23] and Hájek [9]. We also give a new proof of independence of \(B\Sigma_{n+1}\) over \(I\Sigma_n\) by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8].

Mathematics Subject Classification (1991): Primary: 03F30. Secondary: 03F05, 03D20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Lev D. Beklemishev
    • 1
  1. 1. Steklov Mathematical Institute, Gubkina 8, 117966 Moscow, Russia (e-mail: RU

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