Analysis and Extension of Omega-Rule

Abstract

$Ω$-rule was introduced by W. Buchholz to give an ordinal-free cut-elimination proof for a subsystem of analysis with $Π^{1}_{1}$-comprehension. His proof provides cut-free derivations by familiar rules only for arithmetical sequents. When second-order quantifiers are present, they are introduced by $Ω$-rule and some residual cuts are not eliminated. Using an extension of $Ω$-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 $ω$-rule). W. Buchholz used $Ω$-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 $Ω$-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 $Ω$-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 $Π^{1}_{1}$-CA (of the same strength as $ID_{1}$) that W. Buchholz used for his explanation of finite reductions. Extension to full $Π^{1}_{1}$-CA is forthcoming in another paper.