Effectively retractable theories and degrees of undecidability

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.