### The jump is definable in the structure of the degrees of unsolvability

S. Barry Cooper
1990 Bulletin of the American Mathematical Society
Recursion theory deals with computability on the natural numbers. A function ƒ from N to N is computable (or recursive) if it can be calculated by some program on a Turing machine, or equivalently on any other general purpose computer. A major topic of interest, introduced in Post [23], is the notion of relative difficulty of computation. A function ƒ is computable relative to a function g if after equipping the machine with a black box subroutine that provides the values of g, there is a
more » ... m (which now may call g via the subroutine) which computes ƒ. In this case we write ƒ < T g. Two functions are Turing equivalent if each is computable relative to the other; the equivalence classes are called Turing degrees. These degrees form a partial ordering 3 under the induced reducibility relation < . The structural analysis of the partial ordering 3 has been a major area of research in recursion theory since the pioneering paper of Kleene and Post [14] . Kleene and Post proved a number of results on the structure of 3 including the embeddability of arbitrary countable partial orders into 3, and obtained partial results on extendability of a given embedding to a larger domain. This line of investigation was pursued by many people over the next twenty-five years, culminating in essentially complete solutions of these problems, and a characterization of the possible ideals of the structure 3 (see Lachlan and Lebeuf [16] and Lerman [17], [18]). Kleene and Post also considered the enriched structure 3 1 equipped with the "jump operator", denoted ', which is a canonical operation on degrees which takes each degree d to a strictly