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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

1. Cooper, S. B., Minimal degrees and the jump operator, J. Symbolic Logic 38 (1973), 249–271.Google Scholar
2. Epstein, R., Minimal degrees of unsolvability and the full approximation construction, Memoirs Amer. Math. Soc. 162 (1975).Google Scholar
3. Epstein, R. and Posner, D., Diagonalization in degree constructions, to appear, J. Symbolic Logic.Google Scholar
4. Feferman, S., Some applications of the notion of forcing and generic sets, Fund. Math. 55 (1965), 325–345.Google Scholar
5. Jockusch, C., Degrees in which the recursive sets are uniformly recursive, Can. J. Math. 24 (1972), 1092–1099.Google Scholar
6. Lachlan, A. H., Lower bounds for pairs of recursively enumerable degrees, Proc. London Math. Soc. (3) 16 (1966), 537–569.Google Scholar
7. Posner, D., High degrees, Doctoral Dissertation, University of California, Berkeley, 1977.Google Scholar
8. Posner, D. and Robinson, R. W., Degrees joining to 0’ (tentative title), in preparation.Google Scholar
9. Sacks, G. E., A minimal degree less than 0', Bull. Amer. Math. Soc. 67 (1961), 416–419.Google Scholar
10. Sacks, G. E. Degrees of uns olv ability, Ann. of Math. Studies No. 55 (Princeton Univ. Press, Princeton, N.J., 1963).Google Scholar
11. Sasso, L. P., A minimal degree not realizing least possible jump, J. Symbolic Logic 39 (1974), 571–574.Google Scholar
12. Shoenfield, J. R., Degrees of un solvability (North Holland Publishing Company, Amsterdam- London, 1971).Google Scholar
13. Yates, C. E. M., Prioric games and minimal degrees below 0(1), Fund. Math. 82 (1974), 217–237.Google Scholar