§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions ; and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his (...) thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a. (shrink)

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years.

We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.

We show, however, that this is not always the case.

Continuity is perhaps the most familiar characterization of the finitary character of the operations performed in computation. We sketch the historical and conceptual development of this notion by interpreting it as a unifying theme across three main varieties of semantical theories of programming: denotational, axiomatic and event-based. Our exploration spans the development of this notion from its origins in recursion theory to the forms it takes in the context of the more recent event-based analyses of sequential and concurrent computations, touching (...) upon the relations of continuity with non-determinism. (shrink)