Richard A. Shore. Determining automorphisms of the recursively enumerable sets. Proceedings of the American Mathematical Society, vol. 65 (1977), pp. 318– 325. - Richard A. Shore. The homogeneity conjecture. Proceedings of the National Academy of Sciences of the United States of America, vol. 76 (1979), pp. 4218– 4219. - Richard A. Shore. On homogeneity and definability in the first-order theory of the Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 8– 16. - Richard A. Shore. The arithmetic and Turing degrees are not elementarily equivalent. Archiv für mathematische Logik und Grundlagenforschung, vol. 24 (1984), pp. 137– 139. - Richard A. Shore. The structure of the degrees of unsolvabitity. Recursion theory, edited by Anil Nerode and Richard A. Shore, Proceedings of symposia in pure mathematics, vol. 42, American Mathematical Society, Providence1985, pp. 33– 51. - Theodore A. Slaman and W. Hugh Woodin. Definability in the Turing degrees. Illinois journal of mathematics, vol. 30 (1986), pp. 320– 334.

1990 ◽  
Vol 55 (1) ◽  
pp. 358-360
Author(s):  
Carl Jockusch
Keyword(s):  
American Mathematical Society ◽  
The United States ◽  
Recursion Theory ◽  
Order Theory ◽  
Mathematical Society ◽  
Turing Degrees ◽  
National Academy Of Sciences ◽  
First Order Theory ◽  
Pure Mathematics ◽  
Academy Of Sciences

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

1982 ◽  
Vol 47 (1) ◽  
pp. 8-16 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Recursion Theory ◽  
Order Theory ◽  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Recursive Functions ◽  
First Order Theory ◽  
Partial Recursive Functions ◽  
Carry Over

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).

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