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

Published online by Cambridge University Press:  12 March 2014

Mitsuhiro Okada
Mitsuhiro Okada*
Affiliation:
Department of Computer Science, University of Illinois, Urbana, Illinois 61801 Department of Philosophy, Keio University, Tokyo, Japan
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 577 - 581
Copyright
Copyright © Association for Symbolic Logic 1987

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References

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