ABSTRACT
Much of modern computability theory is concerned with understanding the computational complexity of sets of positive integers, yet even in the original paper of Turing [64], a central topic is interest in effectiveness considerations for reals. Of particular interest to computable analysis (e.g., Weihrauch [66], Pour-El [52], Pour-El and Richards [53], Ko [39]) and to algorithmic informa tion theory (e.g., Chaitin [10], Calude [6], Martin-Lof [51], Li-Vitanyi [48]) is the collection of computably enumerable reals. These are the reals a such that the lower cut L(a) consisting of rationals less than a forms a computably enumerable set.
