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An extension of the omega-rule

An extension of the omega-rule The $$\Omega $$ Ω -rule was introduced by W. Buchholz to give an ordinal-free proof of cut-elimination for a subsystem of analysis with $$\Pi ^{1}_{1}$$ Π 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 $$\Omega $$ Ω -rule and some residual cuts are not eliminated. In the present paper, we introduce an extension of the $$\Omega $$ Ω -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 $$\omega $$ ω -rule). In fact we treat the subsystem of $$\Pi ^{1}_{1}$$ Π 1 1 -CA (of the same strength as $$ID_{1}$$ I D 1 ) that W. Buchholz used for his explanation of G. Takeuti’s finite reductions. Extension to full $$\Pi ^{1}_{1}$$ Π 1 1 -CA is planned for another paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

An extension of the omega-rule

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-016-0482-y
Publisher site
See Article on Publisher Site

Abstract

The $$\Omega $$ Ω -rule was introduced by W. Buchholz to give an ordinal-free proof of cut-elimination for a subsystem of analysis with $$\Pi ^{1}_{1}$$ Π 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 $$\Omega $$ Ω -rule and some residual cuts are not eliminated. In the present paper, we introduce an extension of the $$\Omega $$ Ω -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 $$\omega $$ ω -rule). In fact we treat the subsystem of $$\Pi ^{1}_{1}$$ Π 1 1 -CA (of the same strength as $$ID_{1}$$ I D 1 ) that W. Buchholz used for his explanation of G. Takeuti’s finite reductions. Extension to full $$\Pi ^{1}_{1}$$ Π 1 1 -CA is planned for another paper.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Mar 31, 2016

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