INTERPOLATING D-RE AND REA DEGREES BETWEEN RE DEGREES

We provide three new results about interpolating 2-r.e. (i.e. d-r.e.) or 2-REA (recursively enumerable in and above) degrees between given r .e. degrees: Proposition 1.13. If c < h are r.e., c is low and h is hi gh, then there is an a < h which is REA in c but not r.e. Theorem 2.1. For all high r.e. degrees h < g there is a properly d-r.e. degree a s uch that h < a < g and a is r.e. in h. Theorem 3.1. There is an incomp lete nonrecursive r.e. A such that every set REA in A and recursive in 0' is of r.e. degree. The first proof is a variation on the construct ion of Soare and Stob (1982), The second combines highness with a modi fied version of the proof strategy of Cooper et al. (1989). The third theorem is a rather surprising result with a somewhat unusual proof st rategy. Its proof is a 0''' argument that at times moves left in the t ree so that the accessible nodes are not linearly ordered at each stag e, Thus the construction lacks a true path in the usual sense. Two sub stitute notions fill this role: The true nodes are the leftmost ones a ccessible infinitely often; the semitrue nodes are the leftmost ones s uch that there are infinitely many stages at which some extension is a ccessible. Another unusual feature of the construction is that it invo lves using distinct priority orderings to control the interactions of different parts of the construction.