The Isolated D. R. E. Degrees are Dense in the R. E. Degrees

A Turing degree a is said to isolate another degree b in the r.e. degrees if and only if a < b and there is no r.e. degree c with a < c < b. In this case b is called an isolated degree and a an isolating degree. There are of course analogous deenitions for any subsets A and B of !. If a isolates b, it follows from the Sacks density theorem that at least one of the two must fail to be r.e. When the lower degree a is r.e. and the upper one b is not, the deenition of isolation is equivalent to the apparently stronger condition that for every r.e. c b we also have c a. For if c is any r.e. degree below b and a is an r.e. degree which isolates b, then, since c a is r.e. and b is not, we have a c a < b. Since a isolates b, a = c a, so c a. If the isolated degree b is actually r.e. in the isolating r.e. degree a, then b is a 2-REA degree with a maximal r.e. degree below it in which it is r.e. The simplest such case occurs when b is actually a d. sets and h is any one-to-one onto recursive function from ! to W, then it is straightforward to show B is recursively enumerable in and above h ?1 V T W]. While the degree of h ?1 V T W] depends on the particular representation using W and V , it is independent of the enumerating function h, so by a slight abuse of notation we write ~ B for h ?1 V T W] whenever some representation B = W ?V is understood, taking the enumerating function h to be whatever seems most natural from the context. Cooper and Yi ((2]) showed that isolated d.r.e degrees exist by explicitly constructing an isolating r.e. set along with the isolated d.r.e. set. However it is in fact possible to construct an isolated d.r.e. set D so that ~ D isolates it. The simpliications introduced by this method make it possible to combine the construction with a technique for proving weak density results due to Cooper, Lempp, and Watson ((1], Theorem 1), and thereby to show the density of the isolated d.r.e. degrees in the r.e. degrees. Theorem 1 below is …