Archive for Mathematical Logic

, Volume 27, Issue 1, pp 51–60

# Induktive Definitionen und Dilatoren

• Wilfried Buchholz
Article

# 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).

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### Literatur

1. 1.
Girard, J.Y.: A survey ofΠ 21-logic. In: Barwise, J., Kaplan, D., Keisler, H.J., Suppes, P., Troelstra, A.S. (eds.), Logic, methodology and philosophy of science. VI. pp. 89–107. Amsterdam: North-Holland 1982Google Scholar
2. 2.
Girard, J.Y.:Π 21-logic, Part 1: Dilators. Ann. Math. Logic21, 75–219 (1981)Google Scholar
3. 3.
Girard, J.Y.: Introduction toΠ 21-logic. Synthese62, 191–216 (1985)Google Scholar
4. 4.
Girard, J.Y., Normann, D.: Set recursion andΠ 21-logic. Ann. Pure Appl. Logic28, 255–286 (1985)Google Scholar
5. 5.
Jäger, G.: Countable admissible ordinals and dilators. Z. Math. Logik Grundlagen Math.32, 451–456 (1986)Google Scholar
6. 6.
Jervell, H.: Introducing homogeneous trees. In: Proc. Herbrand Symposion, Logic Colloquium 1981, pp. 147–158. Amsterdam: North-Holland 1982Google Scholar
7. 7.
Ressayre, J.P.: Bounding generalized recursive functions of ordinals by effective functors; a complement to the Girard theorem. In: Proc. Herbrand Symposion, Logic Colloquium 1981, pp. 251–279. Amsterdam: North-Holland 1982Google Scholar
8. 8.
Van de Wiele, J.: Recursive dilators and generalized recursions. In: Proc. Herbrand Symposion, Logic Colloquium 1981, pp. 325–332. Amsterdam: North-Holland 1982Google Scholar
9. 9.
Päppinghaus, P.: Ptykes in GödelsT und Verallgemeinerte Rekursion über Mengen und Ordinalzahlen. Habilitationsschrift, Hannover 1985Google Scholar
10. 10.
Schütte, K.: Proof theory. Berlin Heidelberg New York: Springer 1977Google Scholar