Turing degrees of certain isomorphic images of computable relations Original Research Article

A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let View the MathML source be a computable model and let R be an extra relation on the domain of View the MathML source. That is, R is not named in the language of View the MathML source. We define View the MathML source to be the set of Turing degrees of the images f(R) under all isomorphisms f from View the MathML source to computable models. We investigate conditions on View the MathML source and R which are sufficient and necessary for View the MathML source to contain every Turing degree. These conditions imply that if every Turing degree ⩽ 0″ can be realized in View the MathML source via an isomorphism of the same Turing degree as its image of R, then View the MathML source contains every Turing degree. We also discuss an example of View the MathML source and R whose View the MathML source coincides with the Turing degrees which are ⩽ 0′.