Banach–Mazur games, comeager sets and degrees of unsolvability

Abstract

This paper provides a framework for studying the general theory of the upper semi-lattice of degrees of unsolvability. A less obvious and more complicated extension of this framework has been initiated in our papers (17) and (18); these are concerned not with the general theory but with the Priority Method, for which a useful formal framework has long been sought. Our purpose here is partly to provide some background for these other papers, which should help explicate their less obvious aspects; but mainly we wish to illustrate the advantages of this framework from a purely expository point of view. There is an additional advantage in that a formal framework makes it possible to pose certain precise questions which were vague before, and even on occasion to answer them.