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An extension of the omega-rule

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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.

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Authors and Affiliations

  1. Waseda Institute for Advanced Study, 1-6-1 Nishi Waseda, Shinjuku-ku, Tokyo, Japan

    Ryota Akiyoshi

  2. Department of Philosophy, Stanford University, Building 90, Stanford, CA, 94305-2155, USA

    Grigori Mints

Authors
  1. Ryota Akiyoshi
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  2. Grigori Mints
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Correspondence to Ryota Akiyoshi.

Additional information

The main result of this paper was obtained in 2011. After Professor Mints’ passing away in 2014, this paper was completed by the first author. He would like to express his deep gratitude to Professor Mints for collaborations including this joint work.

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Akiyoshi, R., Mints, G. An extension of the omega-rule. Arch. Math. Logic 55, 593–603 (2016). https://doi.org/10.1007/s00153-016-0482-y

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