What are good books/other readings for elementary set theory?

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

http://books.google.com/books?id=x6cZBQ9qtgoC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

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?]

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

http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407/ref=pd_sim_b_26

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.

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.

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.