Generic sets and minimal $\alpha $-degrees

C. T. Chong
1979 Transactions of the American Mathematical Society  
A non-a-recursive subset G of an admissible ordinal a is of minimal a-degree if every set of strictly lower a-degree than that of G is a-recursive. We give a characterization of regular sets of minimal a-degree below 0' via the notion of genericity. We then apply this to outline some 'minimum requirements' to be satisfied by any construction of a set of minimal N -degree below 0'. In 1956 Spector [8] showed the existence of a minimal Turing degree. This result stimulated the study of initial
more » ... ments of degrees of unsolvability (cf. Yates [9]), and the technique used in Spector's proof led Sacks to the formulation of the method of forcing with perfect closed sets (cf. Sacks [4]), a method which proved to be very important in higher recursion theory. Despite such success, the basic problem of proving the existence of a minimal a-degree, for all admissible ordinals a, remains only partially answered. The best result to date is Maass's proof [3] that minimal a-degrees exist if the 22-cofinality of a (a2cf(a)) is not less than the 22-projectum of a (o2p(a)). This result improves upon Shore's [6] where 22-admissibility of a was assumed. Nevertheless, the solution for the minimal a-degree problem, in the general case when a is not 22-admissible, remains open. Indeed it is not even known whether minimal N^-degrees exist (N¿ = the wth constructible cardinal
doi:10.1090/s0002-9947-1979-0539912-0 fatcat:ldhiaws3jfhankdstthxe6pdg4