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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.

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    $\begingroup$ Perhaps Arnold Miller's Descriptive Set Theory and Forcing? I assume you know about the texts by Kechris, Moschovakis, and Srivastava, since these have been around a while and don't make use of forcing ideas. $\endgroup$ – Dave L. Renfro Jul 2 '14 at 14:30
  • $\begingroup$ @DaveL.Renfro By more in the spirit of Kunen, I actually meant a text book more intended to be a book for instruction rather more than a reference. Thank you for the links. I will check them out. $\endgroup$ – UserB1234 Jul 2 '14 at 14:36
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    $\begingroup$ @Dave: Miller's book can be found for free on arXiv, by the way. $\endgroup$ – Asaf Karagila Jul 2 '14 at 15:26
  • $\begingroup$ @Asaf: Thank you. It looks very interesting. $\endgroup$ – UserB1234 Jul 2 '14 at 16:16
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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.

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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.

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