Reference request, Descriptive set theory

Question

I was wondering what a good text would be to learn descriptive set theory out of? Hopefully something more in the spirit of Kunen's text on the introduction to independence proofs.

Answer 1

Descriptive set theory has many flavors. Particular texts are more suited for certain directions.

The newer direction is the application of descriptive set theory toward other fields in mathematics. This includes equivalence relation theory, descriptive graph theory, dynamical systems, Polish groups, etc. In addition to a broad knowledge of mathematics, these uses more classical techniques. Classical here means the technique tend to use topology, measure, and category arguments. Kechris Classical Descriptive Set Theory is a good place to learn what is needed for this area.

However, even in the above, effective descriptive set theory, determinacy, and forcing ideas occasionally appear. For example, in equivalence relation theory and descriptive graph theory, the dichotomy theorems were originally proved using effective descriptive set theory (although there now exists classical proofs). Moschovakis Desriptive Set Theory is a good place to learn effective descriptive set theory; although he does not cover the Gandy-Harrington topology. Perhaps the paper on Determinacy and effectively descriptive set theory by Kechris and Martin, the relevant chapters in Jech Set Theory, and the first chapter of Invariant Descriptive Set Theory by Gao would be a good supplement for learning effective descriptive set theory. Some knowledge of recursion theory is neccessary. Kanovei's Borel Equivalence Relation proves some of these dichotomies using Gandy-Harrington topology but in more of a forcing flavor.

For a general overview of descriptive set theory including topics like uniformization, reduction, scales, determinacy, the texts by Kechris and Moschovakis are good.

For the interaction of descriptive set theory with general set theory ideas like constructibility, consistency results, large cardinals, etc. : Moschovakis last chapter has some. The relevant sections in Jech Set Theory and Kanaomori The Higher Infinite may also be helpful. The new book by Ralf Schindler has many of these results as well as well as the proof of consistency of projective determinacy from large cardinals.

Answer 2

I have the same idea that Hjorth about the backgruound on descriptive set theory. One big idea is study the "basic" technics. 1-19 in Classical descriptive set theory. After, one can take a specific way.

Answer 3

Moschovakis' Descriptive Set Theory is the standard text for anyone who wants to learn descriptive set theory. However, this book does not contain

1. The in-depth inner model theory
2. Kechris' application of descriptive set theory to the other areas, including the theory of countable Borel equivalence relations.

Having read Moschovakis' book, if you want to know more about 1, I suggest reading books about Woodin's core model (Note that I've been told this studying involves a very very very long way.) If you want to know more about 2, Kechris' Classical Descriptive Set Theory is the standard book. A lot of the theorems in this book are used frequently when reading the latest paper concerning descriptive set theory. If you want to know more about the countable Borel equivalence relation, Kechris' The Theory of Countable Borel Equivalence Relations is available on his own website. This is a survey of up-to-date progress in the countable Borel equivalence relations.