History of Descriptive Set Theory
How did descriptive set theory come to look the way it does today?
Although the emergence of descriptive set theory is well-documented by Kanamori’s article of the same name, I am curious about the evolution of certain tools, concepts, and practices that have come to shape the subject into what it looks like today. For example, when did people start using ill-founded trees instead of Luzin sieves and constituents as a main tool for structural investigations of the analytic set? When did it become acceptable to freely use metamathematical tools in studying the projective hierarchy?
Currently I’m trying to chronicle the relevant events preceding the consolidation of Borel equivalence relations as a subfield of mathematics. In particular, I trace the development of mathematical tools and problems that motivated the study of Borel equivalence relations, and the development of the field itself.