Recursive enumerability and the jump operator

Abstract

By degree we mean degree of recursive unsolvability as defined by Kleene and Post in [4]. Following Shoenfield [7], we say a degree c is recursively enumerable in a degree if there is a set of degree c which is the range of a function of degree less than or equal to b, and we call a degree recursively enumerable if it is recursively enumerable in 0 (i.e., if it is the degree of a recursively enumerable set). The jump operator, which takes the degree d to the degree d' (the completion of d), was defined in [4] and has the following properties: if h is recursively enumerable in d, then h d; and d' is recursively enumerable in d. In [4] a degree c is said to be complete if there exists a degree d such that d' = c. Friedberg [1] showed that a degree c is complete if and only if c ? 0'. For any degree b, if < d < b', then b' < d' < and d' is recursively enumerable in b'. Shoenfield [7] proved that if b' < c ? b and c is recursively enumerable in b', then there is a degree d such that ? d ? b' and d' = c. Thus the degrees which lie between b' and b and are recursively enumerable in b' can be viewed as the completions of the degrees which lie between and b'. He also showed there is a degree greater than and less than b' which is not recursively enumerable in b. Our main result below is that the degrees which lie between b' and b and are recursively enumerable in b' can be viewed as the completions of the degrees which lie between and b' and are recursively enumerable in b. Our notation is that of [3].