Splitting and jump inversion in the Turing degrees

Abstract

It is shown that for any computably enumerable degree a 6= 0, any degree c 6= 0, and any Turing degree s, if s ≥ 0, and c.e. in a, then there exists a c.e. degree x with the following properties, (1) x < a, c 6≤ x, (2) a is splittable over x, and (3) x = s. This implies that the Sacks’ splitting theorem and the Sacks’ jump theorem can be uniformly combined. A corollary is that there is no atomic jump class consisting entirely of Harrington non-splitting bases.