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Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set theory (viewed from the boldface side of things)? Specifically, I mean theorems which are in Kechris, Moschovakis, Srivastava. I know that a lot of knowledge goes into understanding the "big theorems" and this will have to be obtained in the journey.

My point: I'm working through Kechris and would like some sort of guide posts to help me know where I am in the subject. I would like the theorems to be from these books and not any theorems from current literature.

Also, are there any books that can help me in the subject besides the 3 I mentioned above?

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This question should probably be community wiki, because there is not a unique correct answer. – Trevor Wilson Oct 15 '12 at 3:54
@Trevor, I disagree about this being CW (and the SE 2.0 does not allow the OP to set CW on their own anyway; just for answers). – Asaf Karagila Oct 15 '12 at 8:09

Here are the first few I can think of.

  • The Baire category theorem
  • Mostowski's absoluteness Theorem
  • Shoenfield's absoluteness theorem
  • The Mansfield–Solovay theorem
  • The Martin–Solovay Theorem
  • Martin's proof of analytic determinacy
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I would like to add on Trevor's list:

  • Gale-Stewart theorem (or determinacy of any simple-enough class).
  • Borel determinacy.
  • $\Sigma^1_1$ determinacy implies $0^\#$.
  • Solovay's construction of a model where all sets are Lebesgue measurable (while from a descriptive point of view it is not the most exciting theorem, it is an important proof).

I'm not sure whether or not it's too far, but in a yearly course I'd definitely expect to hear something about it:

  • The relations between Woodin cardinals and Projective Determinacy

It is somewhat unclear to me how far do you want to go with this. Do you want to end up knowing? I would consider the projective determinacy equivalence with Woodin cardinals (and the natural extension of AD with infinitely many of those) quite an advanced theorem, but maybe you would like to go beyond that. Maybe you would like to know stationary tower forcing; and theorems about the absoluteness of the theory of $L(\mathbb R)$. That really depends on you.

As for references, I would also suggest Miller's book:

Descriptive Set Theory and Forcing: how to prove theorems about Borel sets the hard way. Lecture Notes in Logic 4(1995), Springer-Verlag.

Which one can find for free on his homepage or on ArXiv.

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