Priority Arguments in Higher Recursion Theory

Abstract

This is the first of six lectures on priority arguments in higher recursion theory. The term “priority argument” refers to a mode of combinatorial reasoning introduced independently by Friedberg [1] and Muchnik [2] to obtain a positive solution to Post's problem, the existence of two recursively enumerable sets such that neither is recursive in the other. I will try to show how their idea lifts to three generalizations of ordinary recursion theory. The three are: α-recursion, where α is a ∑1 admissible ordinal; β-recursion, where β is little more than a limit ordinal; and Kleene recursion in F, a normal object of type 3. Some of the results to be described are new and, in the case of Kleene recursion, not hitherto announced. Others are old but viewed, it is hoped, in the light of a new day.