MacNeille completion and Buchholz' Omega rule for parameter-free second order logics
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Buchholz' Omega-rule is a way to give a syntactic, possibly ordinal-free, proof of cut elimination for various impredicative systems of arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada's algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the Dedekind-MacNeille completion together with Girard's reducibility candidates. Interestingly, it turns out that the Omega-rule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion. This observation naturally leads to an algebraic form of the Ω-rule that we call the Omega-interpretation, that partly appears in (Altenkirch-Coquand 2001). In this paper, we introduce sequent calculi for the parameter-free fragments of second order intuitionistic logic, and explain how use of reducibility candidates in (Maehara 1991) and (Okada 1996) can be avoided by means of the Omega-interpretation. It results in an algebraic proof of cut elimination formalizable in theories of finitely iterated inductive definitions, that can be compared with a result by (Aehlig 2005).
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