The Ω -rule was introduced by W. Buchholz to give an ordinal-free proof of cut-elimination for a subsystem of analysis with 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 Ω -rule and some residual cuts are not eliminated. In the present paper, we introduce an extension of the 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 ω-rule). In fact we treat the subsystem of Π 1 1-CA (of the same strength as ID1) that W. Buchholz used for his explanation of G. Takeuti’s finite reductions. Extension to full Π 1 1-CA is planned for another paper.
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