A simple relationship between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams

Abstract

Buchholz [4] simplified the system of ordinal notations of the Schütte school (cf. [12]), by using the notion of collapsing functions (cf. [5]). In this paper we give a simple relationship between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams. From this simple relationship it turns out that the structures of these two systems are very close.We give two systems OT(I) (§1) and OT(I, A) (§2) of ordinal notations which are considered generalizations of Buchholz's original system, where I and A are well-ordered sets. The original system OT of Buchholz [4] is OT(ω + 1, {0}) in our sense. Here the set OT(I) of ordinal notations is defined as a subset of the set Od(I) of ordinal diagrams in [6], and the set OT(I, A) of ordinal notations as a subset of the set O(I, A) of ordinal diagrams in [14].