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On many-one degrees1

Published online by Cambridge University Press:  12 March 2014

Shih-Chao Liu
Shih-Chao Liu*
Affiliation:
Institute of Mathematics, Academia Sinica, Taiwan (Formosa), China
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Extract

Two questions were left open in [1]. 1) Do W(σ) and W(r) have the same many-one degree for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? 2) Is W(σ) many-one reducible to W(r) for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? We answered the second question affirmatively in [2]. In this paper we answer the first question in the negative. From the results of [1] [2] and this paper we have a complete picture of the many-one degrees of the predicates e ∈ W(σ)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 2 , June 1963 , pp. 143 - 153
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

This article is intended to form part of the author&s Ph. D. thesis at the University of Wisconsin, under the direction of S. C. Kleene.

References

[1]Kreisel, G., Shoenfield, J., Wang, Hao, Number-theoretic concepts and recursive well-orderings, Archiv für Mathematische Logik und Grundlagenforschung, vol. 5 (1960), pp. 42–64.CrossRefGoogle Scholar
[2]Liu, S. C., A note on many-one reducibility, to appear in this Journal.Google Scholar
[3]Klesne, S. C., Introduction to metamathemattcs, New York and Toronto, Van Nostrana, 1952.Google Scholar