ON THE CE DEGREES REALIZABLE IN ?₁⁰ CLASSES

We study for each computably bounded. Pi(0)(1) class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect Pi(0)(1) class where all c.e. members have degree a. We also show that every Pi(0)(1) set of c.e. indices is realized in some perfect Pi(0)(1) class, and classify the sets of c.e. degrees which can be realized in some.01 class as exactly those with a computable representation.