What is a good text in intermediate set theory?

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.

Solution 1:

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.

Solution 2:

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).

Solution 3:

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.

• Ciesielski: Set theory for the working mathematician See also: http://at.yorku.ca/t/o/p/c/62.htm http://www.math.wvu.edu/~kcies/STbook.html (This book was recommended by Theo, too.)

• Two-volume set Just, Weese: Discovering modern set theory See also: http://www.ohio.edu/people/just/book1.html http://logicmatters.blogspot.com/2009/09/praise-for-just-and-weese.html

Solution 4:

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.