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A note on many·one reducibility

Published online by Cambridge University Press:  12 March 2014

Shih-Chao Liu
Shih-Chao Liu*
Affiliation:
Institute of Mathematics, Academia Sinica, Taiwan (Formosa) China, University of Wisconsin, U.S.A.
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Extract

In this note we partly answer a question left open in [1, p. 52] by proving the following theorem.

Theorem. Supposep0≧2, ωn+1·p0≦σ<ωn+1(p0+1). Then W(σ) is uniformly many-one reducible to W(τ) for τ≧(ωn+1·p0)+1.

To prove this theorem we need only show that there is a recursive function f(e) such that if ∣e∣<σ, then ∣f(e)<(ωn+1·p0)+1, while if ∣e∣ ≮ σ, f(e)∉W. (For the notations see [2].)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 1 , March 1963 , pp. 35 - 42
Copyright
Copyright © Association for Symbolic Logic 1964

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References

[1]Kreisel, G., Shoenfield, J., Wang, H., Number-theoretic concepts and recursive well-orderings, Archive für Mathematische Logik und Grundlagenforschung, vol. 5 (1960). pp. 42–64.CrossRefGoogle Scholar
[2]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163.Google Scholar
[3]Kleene, S. C., Introduction to metamathematics, New York and Toronto, (Van Nostrana), 1952.Google Scholar