Countable initial segments of the degrees of unsolvability

Abstract

In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment S of the degrees be a set of degrees such that, if a, b Є S and a < c < b, then c Є S. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.