Double jumps of minimal degrees

Abstract

This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree a < 0′ satisfies a″ = 0″. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let Hn be the class of degrees a < 0′ such that a(n) = 0(n+1), and let Ln be the class of degrees a ≤ 0′ such that an = 0n. (Observe that Hi ⊆ Lj and Li ⊆ Lj, whenever i ≤ j, and Hi ∩ Lj = ∅ for all i andj.) The result mentioned above may now be restated in the form that every minimal degree a ≤ 0′ is in L2. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree a < 0′ is in H1. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree a < 0′ which is not in L1. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree a < 0′ in L1. Thus each minimal degree a < 0′ lies in exactly one of the two classes L1L2 – L1 and each of the classes contains minimal degrees.Our results are not restricted to the degrees below 0′. We show in fact that every minimal degree a satisfies a″ = (a ∪ 0′)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below 0′ to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let GHn be the class of degrees a such that a(n) = (a ∪ 0′)(n), and let GLn be the class of degrees a such that a(n) = (a ∪ 0′)(n−1).