Intereting Posts

Prove that a continuous image of a closed subset of a compact space is a closed subset
A subring of the field of fractions of a PID is a PID as well.
Largest prime factor of 600851475143
Continued fraction expansion related to exponential generating function
Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
Solving a Linear Diophantine Equation
Real and imaginary part of Gamma function
Visualizations of ordinal numbers
Difficult infinite integral involving a Gaussian, Bessel function and complex singularities
if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$
Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.
let $A$ be any inductive set, then $\{C \in P(A)|C \text{ is inductive set} \}$ is a set? … and $\mathbb{N}$…?
If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well
Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only?

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!

- Good problem book in differential geometry
- What's a good book on advanced linear algebra?
- Geometry Book Recommendation?
- Dummit and Foote as a First Text in Abstract Algebra
- Good books and lecture notes about category theory.
- Looking for a book: $B(H)$ not reflexive

- How to Decompose $\mathbb{N}$ like this?
- The cardinality of $\mathbb{R}/\mathbb Q$
- How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
- $\subset$ vs $\subseteq$ when *not* referring to strict inclusion
- Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
- book for metric spaces
- Book on coordinate transformations
- Notation on proving injectivity of a function $f:A^{B\;\cup\; C}\to A^B\times A^C$
- If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?
- The set of all sets of the universe?

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

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.

- Evaluating the integral with trigonometric integrand
- Non-trivial solutions for cyclotomic polynomials
- Is there a closed form for the infinite product $\prod_{n=0}^{\infty}\bigl(1+{x \over 2^n} \bigr)$
- What is the remainder when polynomial $f(x)$ is divided by $(x+1)(x-3)$ when $f(-1) = -4$ and $f(3) = 2$?
- Are vectors and covectors the same thing?
- Surjective Function from a Cantor Set
- Rearranging infinite series
- Examples of nowhere continuous functions
- Category theory, a branch of abstract algebra?
- Consider $u_t – \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$
- Going down theorem fails
- How to show the series $\displaystyle\sum_{\xi\in\mathbb Z^n}\frac{1}{(1+|\xi|^2)^{s/2}}$ converges if and only if $s>n$?
- CW complex such that action induces action of group ring on cellular chain complex.
- What is a null set?
- How to divide polynomial matrices