A Course in Mathematical Logic

Rating:
Summary: Probably the most comprehensive Course in Mathematical Logic
Review: This is probably one of the most oustanding textbooks on advanced mathematical logic written this side of the century.

But, it is surprising to me how difficult it is to find it in any libarary in USA. Even more surprising, however, is that, after all these years, it is not available at an affordable price. (It would be hasty to suggest that some publishers are motivated by greed than a desire to inform and educate, I am sure there are better reasons -- with the reservations that go with saying this, of course.).

On the cheerful side, I had the good fortune to sit on most of this course given by Moshe Machover in London. He is an outstanding logician and teacher. As a human being he is profound and just. He used to say, with a humorous matter-of-factness, something to the effect that he hoped we were not filled with "malaise" his favourite word.

The coverage is thorough and deep. The book is sufficiently advanced for it to be used as a textbook for a Master Level Introduction in Logic. It was being used in this way at London University (in 1979).

Thus it takes you further than most logic books that seek to teach the same set of topics. After covering early theorems in, say, model theory, it goes on to prove advanced theorems well beyond the standard texts on logic. The book, as I mentioned prepares you, relative to the British system, for an MA degree -- and so perhaps, a pre-amble to MPhil and, presuemeably, PhD levels.

You might need to augment this book with some other books in Topology, to follow some of the topological theorems).

Machover's other logic book, intended for Philosophers is set theory, logic and their limitatins, which locally resembles this book but is on a much more modest scale and certainly does not cover constructive logic. His earliest book was Nonstandard Analysis without tears.

In the class, he would give handouts for proof of some of the exercises. But, if you cannot do them on your own you should consult some of the bibliographic sources. For instance on Non-standard models you might consult Robinson's Non-standard Analysis.

His coverage of Set Theory, Limitative Results (Goedel results etc.) and Intuitionistic logic is fantastic.

Buy it if you can afford it.

Rating:
Summary: This is the book you need as a logic primer
Review: When I was in my third year of graduate school and was deciding to specialize in set theory, I realized that it was time to get some formal training in first-order logic and model theory. At our school, there were no courses in foundations at all, so I had to find the right book and map out a course of study for myself. I sat in on a seminar for more advanced students that was going through Chang and Keisler (model theory); although I now use this as a basic reference, at the time I needed a more systematic treatment of first-order logic before getting into the details of model theory.

One day I discovered this book by Bell and Machover. It was exactly what I needed. The first three chapters are just what you need to get a solid grounding in first-order logic, covering the soundness, completeness and compactness theorems.

As I was working through the book, I noticed that the incompleteness theorems weren't treated till the 7th chapter -- this, along with the fact that the exercises were so much fun kept me pushing forward one chapter at a time.

Chapter 4 deals with Boolean algebras. I'd alreday had quite a bit of general topology before starting this chapter, so when the duality between Boolean algebras and Stone spaces is explored in some detail in the exercises, it was really a blast. A lot of material is covered in this chapter; the authors never tell you that they chose just the material you will need if you go on to study the Boolean algebra approach to forcing. (They actually give you a taste of Boolean-valued models in Chapter 5, but that's it for forcing in this book; however, the first author, Bell, wrote another book called Boolean-Valued Models and Independence Proofs in Set Theory that provides the most complete treatment of the subject available in book form.)

Chapter 5 gives a careful treatment of the basics of model theory (which was what I needed). Chapter 6 then gives a very fun treatment of recursion theory. The main material covers enough to prove the equivalence of recursive and computable functions (and computable functions are treated using a special kind of register machine rather than the usual Turing machines).

Finally with all the groundwork completed, the authors give a marvelous treatment of the incompleteness theorems, including the usual results about undecidability of Peano arithmetic and the undefinability of truth. The authors go on to develop recursion theory a little further in light of the theorems of Chapter 7 -- enough to solve Post's problem by building two incomparable r.e. degrees.

The rest of the book consists of special topics -- intuitionistic logic, set theory, and nonstandard analysis. All this was very good, though I suppose I prefer other treatments of set theory (Kunen or Jech).

The strength of this book is that it doesn't gloss over any details -- and this is very important when you first get into mathematical logic. This should be every graduate student's first course on logic!