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 H n be the class of degrees a < 0′ such that a (n) = 0 (n+1) , and let L n be the class of degrees a ≤ 0′ such that a n = 0 n . (Observe that H i ⊆ L j and L i ⊆ L j , whenever i ≤ j , and H i ∩ L j = ∅ for all i and j. ) The result mentioned above may now be restated in the form that every minimal degree a ≤ 0′ is in L 2 . This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree a < 0′ is in H 1 . In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree a < 0′ which is not in L 1 . Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree a < 0′ in L 1 . Thus each minimal degree a < 0′ lies in exactly one of the two classes L 1 L 2 – L 1 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 GH n be the class of degrees a such that a (n) = (a ∪ 0′) (n) , and let GL n be the class of degrees a such that a (n) = (a ∪ 0′) (n−1) .