Minimal upper bounds for arithmetical degrees

This paper was inspired by Lerman [14] in which he proved various properties of upper bounds for the arithmetical degrees. The degrees which are upper bounds for the arithmetical degrees were first studied by Hodes [5] and Enderton and Putnam [5]. Enderton and Putnam [5] showed that if a is an upper bound for the arithmetical degrees, then a (2) ≥ 0 ( ω ) . This area was further studied by Hodes [5], Knight, Lachlan and Soare [12], Jockusch and Simpson [12], and Sacks [17]. Enderton and Putnam [5] and Sacks [17] have combined to show that 0 ( ω ) is the least degree in { a (2) : a is an upper bound for the arithmetical degrees}. So there seem to be some analogies between the degrees of upper bounds for the arithmetical degrees and the degrees below 0 (2) . But Sacks [17] showed an important difference between these two structures; namely, the Turing jumps of upper bounds for the arithmetical degrees have no least element. In Lerman [14], a systematic investigation of properties of the jumps of upper bounds for the arithmetical degrees was suggested, which could lead to a definition of the jump operator over the elementary theory of the partial ordering of the degrees. Although Cooper [5] found a degree-theoretic definition of the jump operator, Lerman's plan is still interesting. We say a degree a is generic if there is a representative A of a such that A is Cohen generic for the arithmetical sentences. By Jockusch [5], the statement that A is Cohen generic for the arithmetical sentences is equivalent to saying that for any arithmetical set S of binary strings, there is a σ extended by A such that σ is in S or no extension of σ is in S . First, we investigate the relation between generic degrees and upper bounds for the arithmetical degrees. In the case D (≤ 0′), the set of degrees below 0′, it is well known that any nonrecursive r.e. degree bounds a 1-generic degree (Cohen generic for Σ 1 sentences). Jockusch and Ponser [12] showed that any degree a with a ″ > T ( a ∪ 0′)′ bounds a 1-generic degree.