Discrete 𝜔-sequences of index sets

Abstract

We define a discrete ω \omega -sequence of index sets to be a sequence { θ A n } n ≥ 0 {\{ \theta {A_n}\} _{n \geq 0}} , of index sets of classes of recursively enumerable sets, such that for each n, θ A n + 1 \theta {A_{n + 1}} is an immediate successor of θ A n \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 {A^S} is the class of recursively enumerable subsets of S, then θ A S \theta {A^S} is at the bottom of c discrete ω \omega -sequences. It follows that every complete Turing degree contains c discrete ω \omega -sequences.