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I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn't taken a set theory course yet?

Thanks a lot for your suggestions!

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up vote 19 down vote accepted

I recommend Naive Set Theory by Halmos. It's a friendly, thin and fun to read introduction to set theory.

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Halmos's Naive Set Theory, although a classic is too chatty. I think Jech's Introduction to Set Theory or Enderton's book does a much better job than Halmos. – user 170039 Apr 17 at 12:49

I am going to go out on a limb and recommend a more elementary book than (I think) any of the ones others have mentioned.

I claim that as a pure mathematician who is not a set theorist, all the set theory I have ever needed to know I learned from Irving Kaplansky's Set Theory and Metric Spaces. (And, you know, I also enjoyed the part about metric spaces). Kaplansky spent most of his career at the University of Chicago. Although he had left for MSRI by the time I got there in the mid 1990's, nevertheless his text was still used for the one undergraduate set theory course they offered there. (Not that I actually took that course, but I digress...)

In fact I think that if you work through this book carefully -- it's beautifully written and reads easily, but is not always as innocuous as it appears -- you will actually come out with more set theory than the average pure mathematician knows.

Apologies if you actually do need or want to know some more serious stuff: there's nothing about, say, cofinalities in there, let alone forcing and whatever else comes later on. But maybe this answer will be appropriate for someone else, if not for you.

Added: I suppose I might as well mention my own lecture notes, available online here (scroll down to Set Theory). I think it is fair to say that these are a digest version of Kaplansky's book, even though they were for the most part not written with that book in hand. [However, last week David Speyer emailed me to kindly point out that I had completely screwed up (not his words!) one of the proofs. He also suggested the correct fix, but I didn't feel sanguine about it until I went back to Kaplansky to see how he did it.]

The description All the set theory I have ever needed to know on the main page is not meant to be offensive to set theorists (and I hope it isn't) but rather an honest admission: here is the little bit of material that goes a very long way indeed. Note especially the word need: this is not to say that these 40 pages contain all the set theory I want to know. For instance, I own Cohen's book on forcing and the Continuum Hypothesis, and I would certainly like to know how that stuff goes...

[Come to think of it: I would be highly amused and interested to read 40 pages of notes entitled All the number theory I have ever needed to know written by one of the several eminent set theorists / logicians who frequent this site and MO. What would make the cut?]

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Enderton's book should be a gentle, easy read for an undergraduate

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+1 for Enderton.This is the book I cut mny teeth on under Russell Miller at Queens. What's most impressive to me about Enderton is how he manages to keep set theory connected to the rest of mathematics while still giving a substantial presentation of ZFC set theory. It's a crime how expensive the book is,that's my only complaint with it. – Mathemagician1234 Sep 12 '11 at 7:07

Halmos's book mentioned above is very gentle and easy, and you should look there first. Afterwards, when I was an undergraduate I remember learning a lot from Set Theory for the Working Mathematician by Krzysztof Ciesielski. In particular, it has a long chapter showing how transfinite induction can be used to construct all sorts of odd subsets of $\mathbb{R}^n$ and what not.

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+1 for Ciesielski;outstanding reading for anyone interested in set theory beyond the basics who aren't very concerned about metamathematics. – Mathemagician1234 Sep 12 '11 at 7:13

I recently bought the book Basic Set Theory by A. Shen and N.K. Vereshchagin and it has been a really nice read. It is very accessible and has a lot of exercises. It covers the basics and is very short, about a 100 pages or so. I would recommend it sincerely, although I'm not sure if it will be too basic for what you already know.

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I just skimmed the table of contents of this book: it seems to be at about the same level as Kaplansky's book (and my lecture notes, for that matter) but written a little more expansively, as is the modern pedagogical style. I have looked through other books in the rather new AMS series "Student Mathematical Library" and as a collection they seem to be very nice. It's too bad they weren't around when I was a (relatively new) student of mathematics. – Pete L. Clark Dec 27 '10 at 12:28
@Pete You're absolutely right about the AMS's "Student Mathematical Library" series. I happen to own two other books in the series, a couple of translations from german, one is Plane Algebraic Curves by Gerd Fischer and the other one is Klaus Hulek's Elementary Algebraic Geometry. Both are really nice although I prefer Hulek's more algebraic approach. – Adrián Barquero Dec 27 '10 at 21:00
@Pete,Adrian Also in this remarkable series of textbooks are John McCleary's A FIRST COURSE IN TOPOLOGY,Richard Palias' differential equations text and several other terrific textbooks that escape me at the moment. It's one of the best textbook series around right now and it deserves the attention of anyone planning to teach serious undergraduate mathematics majors. – Mathemagician1234 Sep 12 '11 at 7:12

I've looked at many set theory books and I think the best one for a beginner is one called Classic Set Theory by Goldrei. It goes through loads of examples and is designed for self-study.

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