This book is an introduction to mathematical logic, covering the syntax and semantics of propositional and first-order logic, the Hilbert-style proof system and its completeness, some model-theoretic material, and Godel's (first) incompleteness theorem. Its more formal and rigorous than most introductory books, which is the style I prefer, but I was left feeling unsatisfied with the book. It was hard to nail down exactly what I didn't like about it, but what I came up with is this: although the theorems and proofs are ok, considered one at a time, the overall perspective of what's going on and how things relate to each other was left hazy. Perhaps better exposition and historical background would correct this, but I found the book unsuitable for self-study for a beginner. This was where I first learned Godel's incompleteness theorem, and even though the version presented is particularly weak (Peano arithmetic is incomplete), I was left confused about the significance of the theorem and exactly what assumptions were used in the course of the proof. I see now that their attempt at simplification is what led to my misunderstandings. If you're looking for a good general mathematical logic book, I seriously recommend you get Enderton instead (see my reviews). If you want a book focused on the incompleteness theorems get Smullyan's excellent GIT.
