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A note on universal sets

Published online by Cambridge University Press:  12 March 2014

A. H. Lachlan
A. H. Lachlan*
Affiliation:
Simon Fraser University
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Extract

In this note is proved the following:

Theorem. Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.

Let α, τ be 1-argument recursive functions such that x goes to (σ(χ), τ(χ)) is a (1–1) map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is called universal if every r.e. set is (many-one) reducible to A; A × B is called universal if the set

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 4 , December 1966 , pp. 573 - 574
Copyright
Copyright © Association for Symbolic Logic 1997

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References

[1]Smullyan, R. M., Theory of Formal Systems, Annals of Mathematics Studies, No. 47, Princeton University Press, Princeton, N.J., 1961.CrossRefGoogle Scholar