Degrees of unsolvability associated with classes of formalized theories

Abstract

In his well-known paper [11], Post founded a general theory of recursively enumerable sets, which had its metamathematical source in questions about the decision problem for deducibility in formal systems. However, in centering attention on the notion of degree of unsolvability, Post set a course for his theory which has rarely returned to this source. Among exceptions to this tendency we may mention, as being closest to the problems considered here, the work of Kleene in [8] pp. 298–316, of Myhill in [10], and of Uspenskij in [15]. It is the purpose of this paper to make some further contributions towards bridging this gap.From a certain point of view, it may be argued that there is no real separation between metamathematics and the theory of recursively enumerable sets. For, if the notion of formal system is construed in a sufficiently wide sense, by taking as ‘axioms’ certain effectively found members of a set of ‘formal objects’ and as ‘proofs’ certain effectively found sequences of these objects, then the set of ‘provable statements’ of such a system may be identified, via Gödel's numbering technique, with a recursively enumerable set; and conversely, each recursively enumerable set is identified in this manner with some formal system (cf. [8] pp. 299–300 and 306). However, the pertinence of Post's theory is no longer clear when we turn to systems formalized within the more conventional framework of the first-order predicate calculus. It is just this restriction which serves to clarify the difference in spirit of the two disciplines.