Properties of the jump classes

On the Turing degree structure we consider an ordering relation, which is that inherited by the Turing reducibility on sets of natural numbers, and also the jump function, which arises naturally as the function on degrees induced by the operator which takes each set to its halting problem. In order to understand the structure it seems a basic task to understand the relationship between these two objects. The aim of this paper is to advocate a line of research which looks to better understand this relationship by establishing the order theoretic properties of degrees in each of the various jump classes. We shall survey a range of results in the area and detail a number of open questions. Some of these questions have been noted in the existing literature, while many seem not to have been previously addressed and may well be amenable to attack by known techniques. For the degrees below 0′, the jump hierarchy was originally introduced by Cooper in [BC] and Soare in [RS].