Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject?

14$\begingroup$ The OP should provide more details about his background and the kind of book he's looking for. $\endgroup$ – Qiaochu Yuan Jul 23 '10 at 18:52

6$\begingroup$ Well, I'm certainly glad that this ticking time bomb of a question which sat here undisturbed for full seven years has finally been outed as offtopic, and its threat to MO safely put to rest. $\endgroup$ – Gerry Myerson Aug 1 '17 at 23:37
Here are a few suggestions (which depending on your background may be more or less useful):
 Logic and Structure by Dirk van Dalen. I have used this as a textbook when teaching mathematical logic and for that purpose it is decent. Some people find it a bit dry, but at least it covers a large amount of material in a reasonably clear manner.
 Mathematical Logic by Joseph R. Shoenfield. This book is, I think, regarded by many logicians as being the gold standard text on the subject.
 A Course in Mathematical Logic by John Bell and Moshe Machover. This is my personal favorite textbook in mathematical logic. (Unfortunately, it's a North Holland book and so is a bit less affordable.)
 A Course in Mathematical Logic for Mathematicians by Yuri I. Manin (with contributions from Boris Zilber). I think that pretty much anything written by Manin is worth taking seriously and this book is no exception.
 Notes on Logic and Set Theory by Peter T. Johnstone. This is a delightful little (literally) book on logic which is highly recommended (perhaps in conjunction with one of the other larger books from this list).
 The Mathematics of Metamathematics by Helena Rasiowa and Roman Sikorski. This is a nice book which gives a lattice theoretic development of mathematical logic. (Difficult to find, but worth a look if your library has a copy.)
 Introduction to Metamathematics by Stephen C. Kleene. A classic text in mathematical logic which is still a rewarding read.
I hope these (admittedly biased) suggestions are some use!

$\begingroup$ All excellent suggestions,Michealespecially Shoenfield and Manin. But how can you leave the classic by my old teacher,INTRODUCTION TO MATHEMATICAL LOGIC by Elliott Mendelson,off any such list?!? Still the best graduate level introduction to the subject,period. $\endgroup$ – The Mathemagician Jul 23 '10 at 16:18

$\begingroup$ As I say, it's a somewhat biased list and I'm sure there are other excellent books (such as Mendelson's) which I forgot to include. $\endgroup$ – Michael A Warren Jul 23 '10 at 16:54

$\begingroup$ Agreed, I think Van Dalen's book is the best introduction to the subject I've seen. $\endgroup$ – Brendan Cordy Jul 23 '10 at 17:09

$\begingroup$ Mendelson is really nice, and although it covers lots of topics (which makes it a nice graduate level introduction), it is also readable, even by some first year undergrads. van Dalen's book is also good, specially its use of natural deduction suits well beginners. Kleene is of course a classic. I would like to add "Mathematical Logic" by J.D. Monk, "A Mathematical Introduction to Logic" by H.B. Enderton, and "Logic for Computer Science" by Jean Gallier. $\endgroup$ – Kaveh Jul 24 '10 at 23:55

$\begingroup$ Thank everyone who provided so many good references. Luckily I found most of your recommendations in the library. I think I will start with Dirk van Dalen's book. :) $\endgroup$ – Yujia Qiu Jul 28 '10 at 10:16
I'm surprised my favorite introduction to mathematical logic hasn't been mentioned by anyone. It's Robert S.Wolf's A Tour Through Mathematical Logic. Wolf has written a book that is extremely compelling to read. His passion for the subject comes through in every sentence. It reads like a novel on mathematial logic and set theory, complete with detailed historical notes, philosophical insights and lots of problems. The book is practically a meditation on the answer to any frustrated student of logic's question,"Why is this important?" I wholeheartedly recommend it as your starting point before you look at any of the moredepth treatments recommended below. As a follow up, I recommend the classic introduction by my old teacher, Elliott Mendelson, An Introduction To Mathematical Logic, a deep and masterfully written introduction for graduate students. Those would be my recommendations.

4$\begingroup$ For "anyone"? Really? Have you asked everyone in the world for their opinion on every introduction to mathematical logic? $\endgroup$ – Qiaochu Yuan Jul 23 '10 at 16:48

6$\begingroup$ 1 for the absurd levels of hyperbole in this answer. Tone it down and I'll remove this downvote. $\endgroup$ – Andy Putman Jul 23 '10 at 17:00

13$\begingroup$ @Andrew L : I removed the downvote. I'm not "punishing" you for writing something nice about a book you like. Rather, I'm trying to convince you to write in a style that conforms to the usual professional standards of mathematics. This style avoids hyperbole, sticks to the facts, and avoids unnecessary personal digressions. I suspect that people would receive your opinions better if you wrote in this style. I suppose you have to decide what is more important to you : "expressing yourself" or getting people to take your opinions seriously. $\endgroup$ – Andy Putman Jul 23 '10 at 18:28

5$\begingroup$ @Andrew: you are free to say "I love this book"; these are statements about your response to it. What I am annoyed by are your statements about other people's responses to it. You can't predict how other people respond to the book  maybe some people will find it too hard, or too easy. You also can't overlook the possibility that you might someday find a book you like even better and/or that is even better suited for the OP. $\endgroup$ – Qiaochu Yuan Jul 23 '10 at 18:39

7$\begingroup$ I second the choice of Wolf's book. It is more current than most of the other suggestions, and it integrates set theory with logic, which is important IMO. I also recommend Mendelson, if only for his appendix with the GentztenSchuette proof of the consistency of arithmetic by transfinite induction. (Very few logic books include this.) $\endgroup$ – John Stillwell Jul 24 '10 at 2:32
Joe Mileti wrote a really nice set of course notes on mathematical logic (approx 20 weeks of lectures). It's a draft for a book titled, I think, "Mathematical Logic for Mathematicians." The course notes are beautifully written (and beautifully delivered, if you've had the chance to see him lecture). He's a very nice guy, and I would suggest contacting him about it.
My memory is a bit hazy about the topics he covered, but we discussed propositional and first order logic, nonstandard analysis, and axiomatic set theory. I also remember that some highlights included connections to graph theory and algebra (I guess this sort of touches upon his research themes).
Set Theory and Logic Robert R. Stoll .
This was our text in some course. See inside. And check your library and/ask the course instructors or try to find any senior students' course outline or better yet check good univ's websites on such a course.
This book is a general textbook on Logic, so it's not for beginners, but anyway, the text is called "Mathematical Logic" by Yu.Ershov and E.A. Palyutin. Sorry I couldn't find a relevant link anywhere. Maybe it's available in some library somewhere?
If you're a beginner to mathematical logic, as you seem to imply, I would strongly recommend you start off by getting acquainted with classical propositional and predicate logic. There is a very useful online set of aritlces on the subject, with interactive exercises. The sections relevant to mathematical logic would be:
 Logic
 Predicate Logic
 Set Theory
 Recursion