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I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to next in set theory? What is a good book for set theory that may be considered 'the next step up'?

If it helps any, my background knowledge consists of some basic abstract algebra, general topology, linear algebra, etc., but I'm not sure how often they are used in real set theory. Thanks.

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up vote 10 down vote accepted

I have not read it myself, however I got a good recommendation from one of my teachers -

Azriel Levy's Basic Set Theory.

Jech's Set Theory is a great book but I think it is indeed slightly too advanced, he writes that the first part contains full proofs (I only read chapters from the second parts, in which proofs are many times sketched out and the details are left for the reader). Once you've got the basic theorems down, one might also check The Handbook Of Set Theory written by an ensemble of competent writers, for more specific topics.

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Agreed, both on the excellence of Levy, and the difficulty of Jech. – Flash Sheridan Mar 9 '11 at 17:43
Thanks Asaf, I flipped through some of Levy today, and it seems quite good. – yunone Mar 10 '11 at 3:28

hmm, well I'm no expert, but you could try Jech's "Set Theory". I haven't read it, but I have glanced through it quite a lot, and it is a huge book, 700+ pages.

It covers basic as well as advanced (majority) set theory (+selected topics) and it starts from zero, although it doesn't teach logic. It is in no way a a foundation of mathematics (like Principia Mathematica), because it is reader friendly.

It is also (illegally) available as an ebook on the net.

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While Jech's book (Leon's recommendation) certainly is an outstanding book, it may be a bit advanced. In my daily work, I find what's covered in Ciesielski's book Set theory for the working mathematician, LMS student texts 32, amply enough for my needs. It makes for an easy read and prepares the reader gently towards forcing. I don't know Enderton's book, but I imagine there's quite a bit of overlap in the beginning. Another book that's often recommended is Kunen's Set theory, an introduction to independence proofs, but I find it a bit hard on the casual reader. Concerning the philosophical background, I found the beginning of Fraenkel, Bar-Hillel, Levy, Foundations of set theory an exciting read (Fraenkel is the F in ZF: Zermelo-Fraenkel).

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Thanks Theo, both those sound promising. – yunone Mar 8 '11 at 10:29

Paul Halmos Naive Set Theory is an Excellent text

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I saw that book listed in a question on good introductory texts, so would you say it's a step up, or about on the same level? If you're not sure, I could try to browse through a copy somehow. – yunone Mar 8 '11 at 10:30
@Yuone: I would like you to have a look into and then decide it for yourself. Thats the best option. – anonymous Mar 8 '11 at 10:32
I'd consider this not a step up from Enderton. In fact, I used it alongside Enderton when I was teaching myself set theory (having minimal mathematical background) and found it much easier than the Enderton. That said, I certainly don't think it is a good intermediate option. – Dennis Nov 2 '13 at 6:48

I've read only some chapters from these books, hopefully enough to be able to give some kind of opinion on them. I think they could be good texts for looking into more advanced set theory.

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