Hierarchies of effective descriptive set theory

Abstract

1. Introduction and summary. The theory of hierarchies deals with the classification of objects according to some measure of their complexity. Such classifications have been fruitful in several areas of mathematics: analysis (descriptive set theory), recursion theory, and the theory of models. Although much of the hierarchy theory of each of these areas was developed independently of the others, Addison, in the series of papers [Ad 1-6], has shown not only that there are deep-seated analogies among these theories, but that indeed many of their results can be derived from those of a general theory of hierarchies. Toward a further consolidation of these theories, this paper will study the relationships and analogies between certain classical hierarchies of descriptive set theory and their counterparts in recursion theory. The roots of modern hierarchy theory lie in the investigations of Baire, Borel, Lebesgue, and others around the turn of the century. As analysts with a concern for the foundations of their subject, they felt that constructions effected by means of the axiom of choice or the set of all countable ordinals were less secure than those carried out by more elementary means. They sought to discover what role these suspect constructions played in analysis and whether or not they could be avoided altogether. Thus descriptive set theory arose with the goal of identifying, classifying, and studying those sets (of real numbers) which were of interest for analysis and for which an construction could be given. Needless to say, there was vigorous disagreement as to just what constituted an explicit construction. The first large class of sets studied were the Borel sets. Since each Borel set can be constructed by iteration of the elementary operations of countable union and