On Degrees of Unsolvability

Abstract

In [4], Kleene and Post showed that there are degrees between 0 (the degree of recursive sets) and 0' (the highest degree of recursively enumerable sets). Friedberg [1] and Muchnik [5] showed that there are recursively enumerable degrees (i.e., degrees containing a recursively enumerable set) between 0 and 0'. The question then arises: are there degrees between 0 and 0' which are not recursively enumerable? Since the recursively enumerable sets can be enumerated by a function of degree 0', the existence of such degrees follows from the following theorem, which may be roughly stated for d = 0 as: if a > 0', then the degrees < a cannot be enumerated by a function of degree < a. DEFINITION. A sequence {a,} of functions is uniformly of degree < a if an(x), as a function of (n, x), is of degree < a.