Pseudojump operators. I. The r.e. case

Abstract

Call an operator J J on the power set of ω \omega a pseudo jump operator if J ( A ) J(A) is uniformly recursively enumerable in A A and A A is recursive in J ( A ) J(A) for all subsets A A of ω \omega . 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 J , every degree ⩾ 0 ′ \geqslant {\mathbf {0}}’ has a representative in the range of J J , and that there is a nonrecursive r.e. set A A with J ( A ) J(A) of degree 0 ′ {\mathbf {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 n {H_n} , L n {L_n} 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 ′ {\mathbf {0}}’ does not split over all lower r.e. degrees.