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Simple Proofs of Some Theorems on High Degrees of Unsolvability

Published online by Cambridge University Press:  20 November 2018

Carl G. Jockusch JR.
Carl G. Jockusch JR.*
Affiliation:
University of Illinois, Urbana, Illinois
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If a is a degree of unsolvability, a is called high if a ≦ 0’ and a’ = 0”. In [1], S. B. Cooper showed that if a is high, then (i) a is not a minimal degree, and (ii) there is a minimal degree b < a. We give new proofs of these results which avoid the intricate priority and recursive approximation arguments of [1] in favor of “oracle” constructions using the recursion theorem. Also our constructions apply to degrees a which are not below 0'. Call a degree ageneralized high if a’ = (a U 0')' . Among the degrees ≦ 0', the generalized high degrees obviously coincide with the high degrees.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 1072 - 1080
Copyright
Copyright © Canadian Mathematical Society 1977

References

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