Customer Reviews

Set Theory (Lecture Notes in Computer Science)

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In 1979, I was a first-year graduate student in mathematics. One summer day, I was looking in the math section of Stanford bookstore and saw this thick green volume with the simple title Set Theory (by Thomas Jech). I couldn't help pulling the tome off the shelf. I flipped through the pages in awe. This book had everything about mathematics that I had always wanted to know.

After about an hour, I reluctantly looked at the price and it was just too much; I had to put it back on the shelf. But for the next month, that book was all I could think about. I finally went back and bought it.

Two years later after hooking up with my adviser and embarking on research in set theory, I started working through Jech's book starting on page 1. It took me 2 years to work through the entire book, and for much of that time I had the opportunity to present what I was learning in seminars.

That book is a real treasure. I don't think I've spent as much time poring over any other book. I think the presentation of material is fantastic and the coverage is thorough (or it was at the time I studied it--probably his recently updated work also has this attribute).

I would recommend this book (or rather the most recent edition of it) to any serious graduate student specializing in set theory.

Two areas where I needed supplementary study were in his approaches to the constructible universe and to forcing. These are important areas, and Jech does a fine job in his approach, but certain approaches other than his have become more of a standard, and any serious researcher will have to become familiar with these standards. Jech uses Boolean algebras (primarily) in his development of forcing (and his development is excellent) whereas by now, the usual approach is with partial orders. Also, Jech develops L as a transitive model that is closed under "Godel operations"--a perfectly valid approach. These days, though, the formula-based approach is more common in the literature.

Nonetheless, Jech's wide variety of forcing applications, his in-depth treatment of large cardinals, and his compact surveys of saturated ideals and descriptive set theory make his work really an outstanding contribution.

The author tried to cover everything a set theorist should master plus a representative selection of topics of current interest. That makes for a lot of ground to cover, but Jech did a great job. The writing is very well organized and clear. Every short chapter has many exercises, often with hints. There are extensive sections on applications of forcing. The indexes are really good.

There has to be a down side, of course. In order to squeeze so much in, he had to be brief. There is little context provided, especially in Part I: Basic Set Theory. There are rarely any examples and only the main facts are covered. That is all part of an understandable compromise, but I have a serious complaint (my only one) about the references. He gives detailed historical references in each chapter, but no references to further reading. He could have done it with hardly any use of space and it would have been very helpful.

Because of the brevity, it is a bit hard to learn from, but it makes a great secondary reference. For example, its explanations are often clearer and more direct than in Kunen and with more detailed proofs. It you are going to have any more exposure to set theory than an introductory course, you will probably want to buy a copy. (BTW, the 2e was just a corrected reprint; 3e is a complete rewrite.)

Just wanted to point out that all the reviews here dated before Feb 2003 are referring to older editions. The new one has been totally revised (no laundry list of corrections at the end) and also expanded -- lots of material from the last 25 years of set theory research is now included. Most notable among these is material on proper forcing and pcf theory. (There is even a section on my research interest, mutually stationary sets, and this is a notion which was just published for the first time 2 years ago!) The book is still just as informative and readable as the previous editions.

EDIT: I still agree with everything I wrote above; nearly 6 years later, I still read portions of this book almost every day (and I'm not even doing set theory per se professionally anymore). However I should state for the record that the book is RIDDLED with typos and minor errors. So, be prepared to read critically.

It's really a good book for researchers in set theory. But it is NOT an introduction for students who want to know what is set theory. You will feel you are so stupid if you read this book without any set theoretical background. I recommand Kunnen's book for those people who are interested in set theory but have no any (or only a little) set theory knowledge.

Unless one is very mature mathematically, I do not think, this is a good first exposure to abstract set theory. I am an engineer by training, with strong mathematical interest. I could barely finish the first part of the book, labelled "Basic Set Theory". It was just too concise, too dense for me.

For the same reason that the book was simply too hard for me, I feel it may be excellent for someone with prior exposure to set theory, and who wants a concise, logically impecable book on the subject.

This is one of the most complete books in set theory. If you are interested in learning set theory (from the basics to research level) this is a must have book.

This book is a wonderful reference volume for set theory. It contains a clear and readable explaination of all the things a set theorist needs to know. I have only one complaint: "revised edition" simply means that a 20-odd page errata has been appended....