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

Abstract

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 program (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

Defining the Turing Jump

“…Cooper [1] argued for the definability of the jump along the lines of the definition of A provided by Jockusch and Shore [3]. We outline the plan of the proof of the definability of A so as to be able to both describe Cooper's proposal and present our own proof, as all of them follow the same general plan.…”

“…Cooper [1] suggested a similar approach to the problem of defining the jump operator. His plan was to use a version of Theorem 1.7 for 2 − r.e.…”

“…For the other direction Cooper [1] claimed as his Main Theorem that there is a suitable 2−r.e. set and so a 2−r.e.…”

“…This would have sufficed to define 0 and so, by relativization, the jump operator itself. Cooper [1] proposed to argue for the existence of such a 2 − r.e. set with a 0 priority construction similar to that proving Lachlan's [7] Nonsplitting Theorem for the recursively enumerable degrees.…”

“…set with a 0 priority construction similar to that proving Lachlan's [7] Nonsplitting Theorem for the recursively enumerable degrees. However, the following theorem shows that there is no such set and so Cooper's [1] proposed property does not define the jump. As Lachlan has proved that every 2 − r.e.…”