This textbook is intended as a one-semester introduction to mathematical logic, covering formal (non-naive) set theory, models, and the theory of proofs (as indicated by the title). The text is broken into four units: (1) an introduction to non-naive set theory with a strong emphasis on the Axiom of Choice and many of its equivalent statements, (2) an introduction to model theory, (3) an introduction to the theory of proof via proof trees, and (4) a concluding chapter on sets that includes the ZFC axioms, Russel’s paradox, formal treatment of cardinal and ordinal numbers, and transfinite induction.

I am trained as a commutative algebraist, but when I was an undergraduate, I took a course on set theory using a text by Hrbacek and Jech. My course spent nearly all of its time on the set theory topics with a little mention of model theory, but I have not studied proof theory before. As a result of my bias, I at first thought the treatment on set theory was slightly too thin. By the end of the text, I had changed my mind given the treatment of models and proof theory in the middle. Overall, the exposition is very clear, and the text is well structured.

Despite great writing from the authors, I suspect that students will struggle somewhat with the model theory chapter. Just make sure to spend adequate time on this chapter as the concepts and notation can be challenging and are also of great importance for the chapter on proof theory. As mentioned, I have never studied proof theory before, but I found this third chapter to be a joy to read and work through. After an introduction to proof trees, the authors eventually work up tot he Completeness Theorem. This theorem is the primary result of the chapter and establishes the subtle connection between the concepts of “a statement is true” versus “a statement can be proven.”

The authors, who are professors at the Utrecht University in The Netherlands, suggest that the material best fits into a third-year undergraduate mathematics curriculum. Although such a suggestion may work well in some settings, I suspect that for many American students, this material will probably need to be saved for the fourth year as there is a heavy reliance on knowledge from abstract vector spaces and ring theory. At the very least, several topics require students to understand equivalence relations and the resulting quotient structures formed by them. Additionally, the prime application of quantifier elimination to broader mathematics given in the text is to algebraic geometry. I thought this application was fascinating as I cannot recall seeing it before in my own study of algebraic geometry. On the other hand, I do not know many American undergraduate mathematics students who would know Hilbert’s Basis Theorem or his Nullstellensatz by their third years in order to appreciate the significance of this application.

My other criticism is a matter of taste. The book includes a large number of exercises that are interspersed within the text. The vast majority are excellent, but (like Atiyah & Macconald’s classic

*Introduction to Commutative Algebra*) many important results are relegated to the exercises when in some cases the text might be clearer if those results were proven in the exposition or given more treatment before jumping to the exercises. As a specific example, the section on induction on proof trees essentially is a statement that such a thing is possible immediately followed by some exercises. The exercises are good, but a couple of sentences explaining to the student what this induction method entails would have helped as the text does not really explain the method, although it provides an example of the method a few pages later.

Beyond those few critiques, I believe this text is very well written and does an excellent job introducing the subject matter to a student. Even if your school does not have a course covering these topics, I would recommend the text for a student conducting an independent study of the material.

Geoffrey Dietz is a Professor of Mathematics at Gannon University in Erie, PA. He is married and has six children.