A General Framework for Simple A\ and 2? Priority Arguments

In this paper an abstraction of simple priority arguments is presented in terms of what are called priorcomeager sets. To illustrate the versatility of the method, a number of different applications are described in §§ 4 and 5. Among them, for example, are the existence of minimal J§ degrees and the nonexistence of minimal 2"? degrees. Priorcomeager sets first appeared in our unpublished lecture notes [11], where separate but similar frameworks for â\ and 2^ arguments were presented. Subsequently, in [12], we have described a framework for the â\ theory in terms of what are called priorie games, connecting these with priorcomeager sets and deducing some additional corollaries. The present common framework for the â\ and 2\ theories is a considerable improvement on these earlier rather inelegant versions. It possesses the additional merit of dealing with some simple priority arguments in the 21 theory below a fixed nonzero IJ degree; these previously needed separate treatment. Finally, it should be mentioned that Lachlan [2] has presented some elegant ideas for a framework which is restricted to the i"? theory but deals with two harder theorems in that theory: the existence of minimal pairs of I\ degrees and the density of the 2\ degrees.