Most mathematicians are, of course, accustomed to dealing with objects like groups and vector spaces. The area of mathematical logic known as model theory refers to objects like these as *structures* and attempts to say intelligent things about them by classifying them according to what statements of first-order logic are true in them. Indeed, a structure is called a *model *of a first-order theory if every statement that defines that theory is true in the structure. For example, the defining conditions of a group can all be expressed as first-order sentences, and so any group is a model of this first-order theory.

Models are ubiquitous in mathematics. It was the discovery of models of hyperbolic geometry in the 19th century that led mathematicians to realize that the Euclidean parallel postulate could not be proved from the remaining axioms of Euclidean geometry; because these models are also expressed in terms of Euclidean geometry, they also established that hyperbolic geometry was at least as consistent as Euclidean geometry — i.e., an inconsistency in hyperbolic geometry would imply the existence of one in Euclidean geometry. And, since Euclidean geometry can be modeled on the real numbers, such an inconsistency would imply one *there* as well. So, models tell us that if hyperbolic geometry were ever shown to be inconsistent, all of mathematics would come tumbling down like a house of cards.

Models, and model theory, also have deeper uses. Tarski proved, for example, that Euclidean geometry (as axiomatized by him) is decidable, in the sense that it is possible to determine, for any statement, whether that statement is true or false. And, at a very sophisticated level, a result in model theory (the Ax-Kochen theorem) bears directly on a purely algebraic conjecture of Emil Artin. (Historical note: many books also credit Ershov as an independent discoverer of this theorem, but the book now under review does not.)

There are other examples of model theory being used to prove algebraic results; for example, Hrushovski gave a model-theoretic proof of the Mordell-Lang conjecture about function fields, and model theory can also be used to prove the Ax-Grothendieck theorem, namely that if a polynomial from complex *n*-space to itself is 1–1, then it is onto. So, although model theory is a branch of mathematical logic, it also has much to offer mathematicians working in other disciplines.

The book under review is an introduction to this branch of mathematics. It covers both the basic terminology and results of the discipline as well as more sophisticated results (such as the aforementioned Ax-Kochen theorem), and also provides interesting applications to other areas of mathematics, notably algebra. Somewhat surprisingly, it does not even presuppose prior completion of a course in mathematical logic; all the material that a reader needs to know about logic is developed in the text, albeit rather rapidly.

The book does, however, assume a strong background in other areas of mathematics, including set theory (including cardinal and ordinal numbers) and abstract algebra (commutative ring theory, field theory, Galois theory, valued fields). Three appendices cover the background material in these areas, but, like many background-review appendices, the speed and succinctness of the exposition makes these more useful for someone reviewing something that he or she has already studied, rather than for someone learning it for the first time.

In view of the fairly sophisticated background material described above, I think that the authors’ statement in the preface that the book is aimed at “lower undergraduate students who would like to work in model theory” is *seriously* unrealistic, even with the astonishing word “lower” removed. I do not know of very many undergraduate students who are comfortable with Galois theory and fields with valuation, or who know very much, if at all, about ordinal numbers. A more appropriate audience, I think, would be first or second year graduate students; perhaps the word “undergraduate” is a typo.

The main body of the text is in seven chapters, the first six of which the authors describe as “the core of model theory which all researchers should start with.” Experts in the area can see what is covered by consulting the table of contents, which is linked above. Suffice it to say now that the book can be thought of as being divided into three parts. The first two chapters comprise a basic introduction to logic and the language and very basic results of model theory. Much of this material (e.g., the compactness theorem, the upward and downward Löwenheim-Skolem theorems) is covered in introductory mathematical logic textbooks such as (to name two of my favorites) Hodel’s *Introduction to Mathematical Logic* and *A Friendly Introduction to Mathematical Logic* by Leary and Kristiansen. The next four chapters are more specialized and “can be termed as the beginning of modern model theory”. The final chapter wades into deeper waters by discussing the Ax-Kochen theorem and its applications to the Artin conjecture.

The authors’ writing style is reasonably clear but quite succinct. My impression is that other books covering similar topics seem to spend more time on exposition. The chapter here on Ax-Kochen is only about 16 pages long; the discussion in, for example, *Mathematical Logic and Model Theory* by Prestel and Delzell takes about 45 pages.

The writing style also strikes me as rather dry. It belongs, I would say, to the Jack Webb “just the facts, ma’am” school of exposition. Things that need to be said, generally *are* said, but the book does not have the chatty, conversational tone of, say, *A Shorter Model Theory* by Hodges. The Prestel and Delzell book cited above, and Marker’s *Model Theory: An Introduction*, also seem to me to have more in the way of expository motivation than does this text. Contrast, for example, the Introduction to Marker’s book with the Introduction to this one: Marker spends several pages describing what model theory is all about, explains the two main kinds of questions one can ask, and explains also how these two main areas are linked via the work of Hrushovski that was referred to above. The book under review spends one paragraph describing in very general terms what model theory is, with no details or specific examples.

I also noticed some odd stylistic quirks. For example, definite articles like “a” and “the” in sentences are often omitted. (Examples: Section 2.3 is titled “Some Consequences of Compactness Theorem”; another sentence in the book reads “In the next chapter, we shall prove so-called upward Löwenheim-Skolem theorem….”). The notation can be quirky as well: in contravention of standard algebraic notation, the symbol + is used to denote the binary operation of an arbitrary (i.e., possibly non-abelian) group.

Nevertheless, for the right audience, this seems like a valuable book. It contains both the basics of the subject as well as more advanced material; a friend of mine (who, unlike me, is an expert in logic) thought that both the selection and organization of topics was good. An instructor of a graduate course in this topic would certainly want to give this book a look, and the inclusion of material like Ax-Kochen and Morley’s categoricity theorem makes this book a useful reference: none of the three model theory books referred to earlier, for example, contain proofs of *both* of these results. (The Hodges text proves the latter but not the former, as does Marker’s book, although Marker does give a detailed description of what Ax-Kochen says. Prestel and Delzell do the opposite: they prove Ax-Kochen and give a detailed statement of Morley’s theorem.)

To summarize and conclude: a useful book to have on one’s shelves, but, as a textbook, it faces some stiff competition from, say, the three other books cited earlier.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.