Dominical categories: recursion theory without elements

Abstract

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developedwithout referencetoelements.By Gödel's generalized incompleteness theorem for consistent arithmetical systemTwe mean any statement of the following sort:(1) if every recursive set is definable inT, thenTis essentially undecidable [41]; or(2) if all recursive functions are definable inT, thenTis essentially undecidable [41]; or(3) if every recursive set is definable inT, thenT0andR0(the sets of Gödel numbers of the theorems and refutables ofT) are recursively inseparable [39]; or(4) if all re sets are representable inT, thenT0is creative [28], [39]; or(5) ifTis a Rosser theory (i.e., all disjoint re sets are strongly separable inT), thenT0andR0are effectively inseparable [39].