Set Theory and Logic

Rating:
Summary: Unusually clear treatment of very abstract matters.
Review: I am not a mathematician. Even so, I can understand, with effort a lot of what the author is trying to say. The choice of topics is outstanding. Excellent coverage of the three way crossroads where logic, modern algebra, and metamathematics intersect.

Rating:
Summary: Very Terse Treatment of a Broad Range of Topics
Review: If you are already familiar with the material, this book is a concise and clear reference, and yes a great buy. But for learning these topics from the beginning, you would be better served by other books that are focused on just a particular topic.

For example, for logic in the context of set theory, I highly recommend Daniel Velleman's How to Prove it.

Rating:
Summary: Good book that falls short of being a great book.
Review: This book is very well written and easy to understand. However, it has a very serious shortcoming: there are no solutions to the exercises. If you're looking for a basic reference, this book is good, but if you want a book you can use to learn set theory and logic, get one that has solutions to the exercises.

Rating:
Summary: Incredible Best Buy
Review: This book is without peer in its breadth of coverage of the foundations of mathematics and logic. I have given this book only 4 stars, because its treatment of any given topic
is not classic. It is the total package that astounds.
For a mere $15, you get a challenging undergraduate introduction to all of the following topics. I have written in parentheses the names of authors of more definitive treatments:

Intuitive set theory through the axiom of choice (Halmos)

Natural numbers Æ Integers Æ Rationals Æ Reals (Feferman)

Mathematical logic (Machover, Smullyan)

Metamathetics (Machover, Mendelson)

Introduction to the axiomatic approach

ZF axiomatic set theory (Suppes)

Boolean algebra through Stone's theorem and the completeness of sentential logic (Halmos & Givant)

Algebra (Birkhoff & MacLane's "Algebra")

Stoll's style is quite discursive, far from the terse lemma-theorem-corollary-remark style of so much 20th century mathematics. My only major disappointment is that the formal proof technique set out in chpt. 4 is natural deduction rather than the tableau method or Quine's Main Method.

It is indeed the case that there are no solutions to the exercises, but I do not believe that that is a major flaw.