On the Degrees of Index Sets

A class of recursively enumerable sets may be classified either as an object in itself — the range of a two-place function in the obvious way — or by means of the corresponding set of indices. The latter approach is not only more precise but also, as we show below, provides an alternative method for solving certain problems on recursively enumerable sets and their degrees of unsolvability. The main result of the present paper is the computation, for every recursively enumerable degree a, of the degree (in fact, isomorphism-type) of the index-set corresponding to the recursively enumerable sets of degree a: its degree is a(3). It follows from a theorem of Sacks [10] that the degrees of such index-sets are exactly those which are ^ 0<3) and recursively enumerable in 0(3). In particular, this proves Rogers' conjecture [9] that the index-set corresponding to 0(1) is of degree 0<4); partial results on this problem have been obtained by Rogers [9] and by Lacombe (unpublished). The most interesting immediate consequence of our result is a different proof of Sacks' theorem [11] that the recursively enumerable degrees are dense. We refer the reader to Kleene [5] and Sacks [10] for our basic terminology and notation. A useful summary of many results which connect degrees with the arithmetical hierarchy is presented in [9], which is a good background to the present paper since without it the latter would not exist. For an assortment of results on classes of recursively enumerable sets the reader is referred to [2]. If e is a number and A is a set, then we define the partial function 0^ by setting :