Mathematical Logic, 2nd Edition

Excellent text for senior undergraduates and beyond (this is not an introduction to logic). You should already be comfortable with rigorous mathematical proofs, including induction. Yes, there is a lot of notation, but it is consistently used and well-motivated. If you do research in this area, you will find that heaps of notation is typical. This will provide a great basis if you plan to do research in logic programming, SAT solvers, machine-checked proofs (via HOL, Coq, etc.), static analysis/formal verification, and so on. With this text, I clearly saw for the first time the difference between semantic and syntactic proofs and what it means for a system to be consistent or complete.

Used this to teach undergraduate logic this past Fall semester. Still good stuff.

The title of this review really says it all. If you are already familiar with the theorems inside, this is probably an excellent way of brushing up on your metalogic. However, if you are a first time learner, the stuff is going to be frustratingly condensed at best and hopelessly opaque at worst.

My fourth logic textbook. Great introduction. What does the picture on the cover mean?

I got this book as promised.

This is probably one of the best introductions to mathematical logic for those with sufficient mathematical maturity. I especially enjoyed the treatment of the completeness theorem for first-order logic (using Henkin's Theorem), and the treatment of Godel's incompleteness theorem, and Trachtenbrachts incompleteness theorem for second-order logic. Compared to other books, this book tends to go light on the notation. If you do not have sufficient math maturity, then you may want to try Smullyan's book on the subject.

This is the worst ML textbook I have ever read! Not so bad in the contents but unfamiliar symbols and style would confuse you. Mendelson and Shoenfield's books are better options instead of this one!

Intended for a one-semester course, it ignores some of the usual topics in a survey course so it can give a deeper treatment of the nature and adequacy of mathematical proofs. It slights number theory, second-order logic, nonstandard analysis, and set theory. There is only enough on recursion and computability to support the main topic, but it goes deeper than usual on limitative results. What it does cover it does very well. Motivation is rich and exercises follow well from the text. Proofs are very clear. Overall, there is much greater coherence in the development of ideas than you usually see in a survey text. While the writing is very good, there is a shortage of definitions, examples, and exercises. Notation is not always clearly introduced and they adopt so many abbreviations it's hard to keep track of what things mean. I also thought that it was not as clear in the second half, maybe due to the multiple authors. Still, I would choose it over Enderton unless you need lots of exercises for class use.

Good book for a good price. Clear explanations and examples. Only possible downside is an overly minimalistic style for notation.