Aspects of the Turing Jump

X ′ is the canonical example of a set which is definable from X but not recursive in X. The Turing degree of X ′ depends only on the Turing degree of X, so the jump induces an increasing function on the Turing degrees D. In this paper, we will discuss two aspects of the jump and its iterations. First, we will show that they are implicitly characterized by general properties of relative definability. Second, we will present the Shore and Slaman [1999] theorem that the function x 7→ x′ is first order definable in the Turing degrees. Finally, we will pose analogous questions about the relation y is recursively enumerable in x and discuss what is known about them. Our discussion will rest on two technical facts, which are generalizations of the following two theorems.