Initial segments of the degrees of size ℵ1

We settle a series of questions first raised by Yates at the Jerusalem (1968) Colloquium on Mathematical Logic by characterizing the initial segments of the degrees of unsolvability of size ℵ₁: Every upper semi-lattice of size ℵ₁ with zero, in which every element has at most countably many predecessors, is isomorphic to an initial segment of the Turing degrees. The second author was partially supported by a grant from the NSF. The research was carried out while he was on sabbatical leave from Cornell University and a Visiting Professor at the Hebrew University, Jersalem. He would like to thank the Hebrew University and in particular the logicians there for their hospitality.