Archive for Mathematical Logic

, Volume 55, Issue 3–4, pp 593–603 | Cite as

An extension of the omega-rule

Article

Abstract

The \(\Omega \)-rule was introduced by W. Buchholz to give an ordinal-free proof of cut-elimination for a subsystem of analysis with \(\Pi ^{1}_{1}\)-comprehension. W. Buchholz’s proof provides cut-free derivations by familiar rules only for arithmetical sequents. When second-order quantifiers are present, they are introduced by the \(\Omega \)-rule and some residual cuts are not eliminated. In the present paper, we introduce an extension of the \(\Omega \)-rule and prove the complete cut-elimination by the same method as W. Buchholz: any derivation of arbitrary sequent is transformed into its cut-free derivation by the standard rules (with induction replaced by the \(\omega \)-rule). In fact we treat the subsystem of \(\Pi ^{1}_{1}\)-CA (of the same strength as \(ID_{1}\)) that W. Buchholz used for his explanation of G. Takeuti’s finite reductions. Extension to full \(\Pi ^{1}_{1}\)-CA is planned for another paper.

Keywords

Cut-elimination Infinitary proof theory Ordinal analysis 

Mathematics Subject Classification

03F05 03F35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Waseda Institute for Advanced StudyShinjuku-kuJapan
  2. 2.Department of PhilosophyStanford UniversityStanfordUSA

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