, Volume 27, Issue 1, pp 51-60

Induktive Definitionen und Dilatoren

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Summary

In this paper we give a new and comparatively simple proof of the following theorem by Girard [1]:

Ifx \({\cal O}\) y \({\cal O}\) ψ(x,y) (where the relationψ is arithmetic and positive in Kleene's \({\cal O}\) ), then there exists a recursive DilatorD such that ∀αωx \({\cal O}\) ∃y∈ \({\cal O}\) ψ(x, y).”

The essential feature of our proof is its very direct definition of the dilatorD. Within a certain infinitary cutfree system of “inductive logic” (which in fact is a modification of Girard's system in [1]) we construct in a uniform way for each ordinalα a derivation Tα of the formula ∀x ∈ \({\cal O}\) y \({\cal O}\) ψ(x, y), and then defineD immediately from the family (Tα)α∈On. Especially we set D(α):=Kleene-Brouwer length of (Tα).