Archive for Mathematical Logic

, Volume 27, Issue 1, pp 51-60

First online:

Induktive Definitionen und Dilatoren

  • Wilfried BuchholzAffiliated withMathematisches Institut der Universität München

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Inductive definitions and dilators


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∈\({\cal O}\) <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 ∈\({\cal O}\) 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).