Archive for Mathematical Logic

, Volume 27, Issue 1, pp 51–60

# Induktive Definitionen und Dilatoren

• Wilfried Buchholz
Article

DOI: 10.1007/BF01625834

Buchholz, W. Arch Math Logic (1988) 27: 51. doi:10.1007/BF01625834

# Inductive definitions and dilators

## 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∈$${\cal O}$$<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).