Summary
In this paper we give a new and comparatively simple proof of the following theorem by Girard [1]:
“If ∀x∈\({\cal O}\)∃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 ∀α≧ω∀x∈\({\cal O}\) <α∃y∈\({\cal O}\) <D(α) 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 ∈\({\cal O}\) <α∃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).
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Buchholz, W. Induktive Definitionen und Dilatoren. Arch Math Logic 27, 51–60 (1988). https://doi.org/10.1007/BF01625834
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DOI: https://doi.org/10.1007/BF01625834