Jump embeddings in the Turing degrees
Peter G. Hinman, Theodore A. Slaman · Journal of Symbolic Logic · 1991
Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure. Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embedding where D is the set of all Turing degrees and