The recursion theorem in abstract partially ordered algebras, such as operative spaces and others, is the most fundamental result of algebraic recursion theory. The primary aim of the present paper is to prove this theorem for iterative operative spaces in full generality. As an intermediate result, a new and rather large class of models of the combinatory logic is obtained.

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: if every recursive set is definable inT, thenTis essentially undecidable [41]; or if all recursive functions are definable inT, thenTis essentially undecidable [41]; or if every recursive set is definable inT, thenT0andR0 are recursively inseparable [39]; or if all re sets are representable inT, thenT0is creative [28], [39]; or ifTis a Rosser theory, thenT0andR0are effectively inseparable [39]. (shrink)

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

The recursion theorem in abstract partially ordered algebras, such as operative spaces and others, is the most fundamental result of algebraic recursion theory. The primary aim of the present paper is to prove this theorem for iterative operative spaces in full generality. As an intermediate result, a new and rather large class of models of the combinatory logic is obtained.