TREES, NORMS AND SCALES

Summary

Introduction

A. In the past few years descriptive set theory has changed from a miscellany to a subject. The main cause has been the study of various determinacy hypotheses – which seems to require a unified, axiomatic approach. This article describes three unifying principles, sets out their relations, and derives from them the best known elementary facts of classical descriptive set theory. (The approach is due primarily to Moschovakis and not all to the author of this paper.)

Sections 1 – 3 consist essentially of one hour talks on each of the notions: trees, norms, scales – the leisure of print allowing the inclusion of a few digressions and, more important, a slightly more abstract approach than is practical in a talk.

Paragraph B of this section fixes notation and some basic terminology. Paragraph C need't be read immediately. It contains a definition or two that will be referred to later, but consists mainly of a chatty example which, though not necessary to the sequel, may help orient the beginner.

B. ℝ, the set of reals, is ω^ω – the set of functions from ω to ω. We're interested in the descriptive set theory of the pointspaces ℝ^m × ωⁿ; and will use x, y, z for pointspaces and x, y, z, … for their elements.