Customer Reviews

Model Theory (Studies in Logic and Foundations of Mathematics Ser. : Vol 73)

5 star:		(3)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

This book was for a while the classic text in model theory, and it still is a good resource for a student in the area. This was the first model theory text I read, and I've always found the proofs to be clear, straightforward, and easy to read. I came with some background in logic, but Chapter 1 covers all the basic logic you need for this book. Chapters 2, 3, 4.1, and 5.1 are still proably as good a place as any to learn the essential core material of the subject.

My biggest reservation about recommending this book is that it was first written in 1973, and it shows. Although this is the third edition of the book, its original structure is still largely the same. The field of model theory has changed a lot since the early 70's. For instance, in 1978 Shelah wrote a famous book that simultaneously answered many open questions in model theory and changed the direction of the whole subject, and the 1990's have seen many new applications to algebra and other areas of pure math. However, these important developments aren't reflected much in this book. The new sections added to this edition aren't exactly on the cutting edge: "Lindstrom's charaterization of first-order logic" was known at least since the early 1980's, and represents a line of research that doesn't seem to have much to do with model theory today; model completeness and nonstandard universes were studied a lot by Abraham Robinson and his colleages -- in the 1950's and 1960's.

Why is it so important to have an up-to-date textbook, since the theorems in this book are surely no less true now than 30 years ago? A really good book should give the student an idea of the current state of the subject, and this book does not. If you only read this book you might think that model theorists were still preoccupied with proving two-cardinal theorems. (Though if for some reason you really like two-cardinal theorems, then this is the book for you!) Here's some other introductory model theory books written from a modern point of view:

Hodges, _A Shorter Model Theory_
Poizat, _A Course in Model Theory_
Marker, _Model Theory: An Introduction_

I've looked at the first two, and they both seem like they would be good books for a beginner. All three of the books cover essentially all the material in Chang and Keisler, except some advanced topics on ultraproducts -- but for almost all applications you don't need ultraproducts anyway, just compactness. All three books also have more emphasis on applications to other areas of math. The last two books contain some more advanced material on stability theory as well.

One final word on the exercises in Chang and Keisler: in response to other reviewers' comments, I think they are comparable in difficulty to the exercises in most other advanced undergrad or beginning grad level math books I've seen. There are a lot of routine exercises, and also a decent number of slightly tricky exercises. And a few are really hard -- some of the double-starred problems are the topics of research papers! But you can just skip these ones.

This book is the only good source that I've found for the applications of model theory in set theory. Model theorists were more interested in those things back then, I guess. Anyway if you want to know about direct limits and ultraproducts (key constructions in set theory), this is the book to buy. It may interest the logician to know that A Shorter Model Theory doesn't even mention direct limits, which are important to use in understanding the consequences of V=L, especially.

Although I am loathe to admit it, I used this book in an undergraduate non-standard analysis seminar over 20 years ago -- it would, therefore, not be surprising if the commentary that it is a bit dated is apt. Chang and Kiesler were, however, big names in the field, and at the time I took the class, this book was one of the few that felt like a textbook. I also remember it as being relatively easy to read. This may have been why, at the time, it was chronically missing from the shelves of our math library -- not checked out, but simply missing. On a recent visit to the same library, the librarian actually remembered the class and that it had used this book -- when he went to locate it on the shelf, it was missing -- he told me that to this day, it is one of the books that routinely disappears from the shelves (as my contribution to the advancement of Foundations, I recently bought a copy to donate to that library -- or perhaps to some impoverished proto-model theorist). This continuing phenomonon of disappearance I take as a continuing tribute and testament to the utility of the text as an introduction.

When C&K came out, it was a major event because it brought together scattered information in a decent graduate introduction. But it doesn't compare well to more recent books. It is relatively slow and important points don't stand out well. It would benefit from a lot more examples. And it is much too focused on ultraproducts. The third edition does very little to update it; it just adds sections 2.4, 2.5, 3.5, 4.4 and a few pages to the Open Problems and Historical Review. Reading C&K along with Marker, Hodges, and Poizat, I felt that C&K was the only one that added nothing interesting to the others. If you really need to read about ultraproducts, Hodges has a good description.

This is the best book on Model Theory I have seen. It's an excellent resource for anyone who is interested in the subject. I found the "historical notes" section in the back of the book especially insightfull.