Simple Proofs of Some Theorems on High Degrees of Unsolvability

Abstract

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.