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Complementation in the Turing degrees

Published online by Cambridge University Press:  12 March 2014

Theodore A. Slaman  and
John R. Steel
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
John R. Steel
Affiliation:
Department of Mathematics, U. C. L. A., Los Angeles, California 90024
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Abstract

Posner [6] has shown, by a nonuniform proof, that every degree has a complement below 0′. We show that a 1-generic complement for each set of degree between 0 and 0′ can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above ∅′. In the second half of the paper, we show that the complementation of the degrees below 0′ does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 160 - 176
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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