Model theory exercises a huge impact on mathematical logic, indeed on mathematics itself, as is already clear by virtue of the nowadays somewhat prosaic example afforded by Euclidean vs. non-Euclidean geometry in the presence of the Euclidean axioms and postulates. Additionally there is the fundamental example of the role played by model theory vis à vis Zermelo-Fraenkel set theory either with the Axiom of Choice or its negation added. Thus, the subject of model theory is of considerable importance, both intrinsically and as a chapter in set theory and logic.
The present AMS publication, Categoricity, is unquestionably intended for logicians, in particular model theorists. The word “categoricity” in meant in the sense delineated by what is described in the book under review as the very genesis of model theory, Morley’s categoricity theorem to the effect that a countable first-order theory is categorical in an uncountable cardinal if and only if it is categorical in all uncountable cardinals. And here we have, by definition, that a theory is categorical in a certain cardinal if it has a(n essentially) unique model of that cardinality. (As always, essential uniqueness means uniqueness up to isomorphism.)
Logicians are concerned with a lot more than first order sentential calculus, however, and so it turns out that one of the avenues pursued in contemporary model theory concerns the problem of extending the aforementioned categoricity to infinitary logic, “where the basic tool of [the] compactness [theorem] fails.” The incredibly prolific Saharon Shelah, one of the few mathematicians whose output might rival that of Erdös, is one of the driving forces behind this campaign, as is evidenced by the fact that the now fundamental notion of abstract elementary class is of his design. Moreover, as the author of the book under review, John T. Baldwin, puts it in his Introduction: “Shelah’s taxonomy of first order theories by the stability classification established the background for most model theoretic researches in the last 35 years. This book lays out some of the developments in extending this analysis to classes that are defined in non-first order ways.”
Categoricity is accordingly not meant for general consumption and Baldwin stresses that “[a] solid graduate course in model theory is an essential prerequisite for this book.” The book is laid out in four parts, comprising twenty-six chapters “that can be covered in a lecture or two.” There are four appendices, the last one titled “Problems.” Says Baldwin: “These problems range from some that should be immediately accessible to major conjectures and methodological issues and areas in need of development. Many of them have been ‘in the air’ for some time.” Obviously Categoricity is intended to bring young logicians to the point of starting research in a very viable area of model theory. It succeeds very well in doing this.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.