On homogeneity and definability in the first-order theory of the Turing degrees

Relativization—the principle that says one can carry over proofs and theorems about partial recursive functions and Turing degrees to functions partial recursive in any given set A and the Turing degrees of sets in which A is recursive—is a pervasive phenomenon in recursion theory. It led H. Rogers, Jr. [15] to ask if, for every degree d, (≥ d), the partial ordering of Turing degrees above d, is isomorphic to all the degrees . We showed in Shore [17] that this homogeneity conjecture is false. More specifically we proved that if, for some n, the degree of Kleene's (the complete set) is recursive in d(n) then ≇ (≤ d). The key ingredient of the proof was a new version of a result from Nerode and Shore [13] (hereafter NS I) that any isomorphism φ: → (≥ d) must be the identity on some cone, i.e., there is an a called the base of the cone such that b ≥ a ⇒ φ(b) = b. This result was combined with information about minimal covers from Jockusch and Soare [8] and Harrington and Kechris [3] to derive a contradiction from the existence of such an isomorphism if deg() ≤ d(n).