Discrete $\omega$-sequences of index sets

Abstract:

We define a discrete $\omega$-sequence of index sets to be a sequence ${\{ \theta {A_n}\} _{n \geq 0}}$, of index sets of classes of recursively enumerable sets, such that for each n, $\theta {A_{n + 1}}$ is an immediate successor of $\theta {A_n}$ in the partial order of degrees of index sets under one-one reducibility. The main result of this paper is that if S is any set to which the complete set K is not Turing-reducible, and ${A^S}$ is the class of recursively enumerable subsets of S, then $\theta {A^S}$ is at the bottom of c discrete $\omega$-sequences. It follows that every complete Turing degree contains c discrete $\omega$-sequences.