Hyperarithmetical Sets

Abstract

The hyperarithmetical sets of natural numbers were introduced (independently) in the early 1950s by Martin Davis, Andrej Mostowski and Stephen Cole Kleene and their study is surely one of the most significant developments in the theory of computability: they have a rich and interesting structure and they have found applications to many areas of mathematics, including inductive definability, higher-type recursion, descriptive set theory and even classical analysis. This article surveys the development of the subject in its formative period from 1950 to 1960, starting with a discussion of its origins and with some brief pointers to later developments. There are few proofs, chosen partly because of the importance of the results but mostly because they illustrate simple, classical methods specific to this area which are not easy to find in the literature, especially in the treatment of uniformity; and these are given in the spirit (if not the letter) of the methods which were available at the time. This is an elementary, expository article and includes an Appendix which summarizes the few basic facts about computability theory that it assumes.