Analysis and Extension of Omega-Rule

Abstract

$\Omega$-rule was introduced by W. Buchholz to give an ordinal-free cut-elimination proof for a subsystem of analysis with $\Pi^{1}_{1}$-comprehension. His proof provides cut-free derivations by familiar rules only for arithmetical sequents. When second-order quantifiers are present, they are introduced by $\Omega$-rule and some residual cuts are not eliminated. Using an extension of $\Omega$-rule we obtain (by the same method as W. Buchholz) complete cut-elimination: any derivation of arbitrary sequent is transformed into its cut-free derivation by the standard rules (with induction replaced by $\omega$-rule). W. Buchholz used $\Omega$-rule to explain how reductions of finite derivations (used by G. Takeuti for subsystems of analysis) are generated by cut-elimination steps applied to derivations with $\Omega$-rule. We show that the same steps generate standard cut-reduction steps for infinitary derivations with familiar standard rules for second-order quantifiers. This provides an analysis of $\Omega$-rule in terms of standard rules and ordinal-free cut-elimination proof for the system with the standard rules for second-order quantifiers. 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 finite reductions. Extension to full $\Pi^{1}_{1}$-CA is forthcoming in another paper.