## Analysis and Extension of Omega-Rule

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Akiyoshi, R.

Mints, G.

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$\Omega$-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.

##### Keywords

Mathematics - Logic, 03F05