Hierarchies of number-theoretic predicates

Abstract

The existence of hierarchies of point sets in analysis has long been familiar from the work of Borel and Lusin. The study of the hierarchies in number theory which we consider here began with a theorem presented to the Society in 1940 and published in [12], These hierarchies have applications in foundational investigations, but we shall be concerned here with the exploration of their structure (using classical logic). We shall survey the previous results from the beginning, and conclude with a few new ones. We have endeavored to make the exposition complete enough so that the layman in this field can get the gist of the arguments without consulting the references.