Pseudojump operators. I. The r.e. case

Call an operator J on the power set of co a pseudo jump operator if J(A) is uniformly recursively enumerable in A and A is recursive in J(A) for all subsets A of c. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator J, every degree > 0' has a representative in the range of J, and that there is a nonrecursive r.e. set A with J(A) of degree 0'. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the H", Ln hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan's result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington's result that 0' does not split over all lower r.e. degrees. 1. We prove some generalizations of theorems about the Turing jump operator (denoted A H A') to theorems about operators of the form A * A ED WA, for an arbitrary fixed Godel number e. (Here A ED B is the recursive join of A and B and WA is the eth set r.e. in A in a fixed standard enumeration.) For instance, Friedberg (see (25, Theorem 4.1)) showed that there is a nonrecursive r.e. set A with A' TK (where K is a complete r.e. set). We prove by a finite injury priority argument similar to Friedberg's that for every e there is a nonrecursive r.e. set A with A ED WA-T K. Now Friedberg's argument relative to an arbitrary oracle B yields a fixed Godel