Archive for Mathematical Logic

, Volume 55, Issue 3, pp 593–603

An extension of the omega-rule


DOI: 10.1007/s00153-016-0482-y

Cite this article as:
Akiyoshi, R. & Mints, G. Arch. Math. Logic (2016) 55: 593. doi:10.1007/s00153-016-0482-y


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.


Cut-elimination Infinitary proof theory Ordinal analysis 

Mathematics Subject Classification

03F05 03F35 

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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