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A Book of Set Theory

Charles C. Pinter
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Glenn Becker
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This is a re-titled revised and corrected reprint of Pinter’s 1971 Set Theory (Addison-Wesley). The new title was presumably chosen to bring it into line with the author’s popular A Book of Abstract Algebra.

Although I am myself a set-theoretic tyro, I feel comfortable enthusiastically recommending Pinter’s book as a first book in set theory for both undergraduates and determined amateurs. The tone is informal without being irritatingly jocular. It does not pander, coddle, or otherwise insult the intelligence. Topics are presented in a sequence that feels natural and does not wander.

All this is not to say the book is an easy read — this is set theory, after all, not Harry Potter. And set theory is a subtle, slippery realm that props up, lies behind and explicates many other sub-fields of mathematics. A reader with no experience with proofs, for example would find the text quite challenging (not that that is a bad thing) and perhaps forbidding. There are lots of proofs.

A Book of Set Theory opens with a discursive Chapter 0, “Historical Introduction,” which swiftly and pleasantly sets the background. The subsequent short but well-stuffed chapters build in complexity, from basics about classes, sets, and functions to transfinite recursion and issues of consistency and independence. The final chapter concludes with a typically swift discussion of Gödel’s theorems.

The short Bibliography has apparently not been updated — which brings me to a less pleasant point: what, precisely, does get updated in an edition like this? Pinter’s Preface says that the last (and most exciting) chapter is new. The Preface and the back cover description claim that the original text has been “corrected”; nevertheless, I tripped over a troubling number of typographical errors. I expect there are far more than the ones I found. Was the text proofread? Where have all the copyeditors gone?

That complaint aside, A Book of Set Theory is both a solid introduction to the topic and a useful reference. It accomplishes a lot for a book of less than 250 pages. To paraphrase Spencer Tracy (Pat and Mike, 1952), there’s “not much meat on [it], but what’s there is cherce.”

Glenn Becker is a staff member at the Harvard-Smithsonian Center for Astrophysics in Cambridge, MA, where he toils in the data archive of the Chandra X-Ray telescope. He is a “reborn astronomy and mathematics fellow traveler” who spent far too many years getting advanced degrees in theater, only to ultimately abandon the entire discipline.

Chapter 1 - Classes and Sets

Chapter 2 - Functions

Chapter 3 - Relations

Chapter 4 - Partially Ordered Classes

Chapter 5 - The Axiom of Choice and Related Principles

Chapter 6 - The Natural Numbers

Chapter 7 - Finite and Infinite Sets

Chapter 8 - Arithmetic of Cardinal Numbers

Chapter 9 - Arithmetic of the Ordinal Numbers

Chapter 10 - Transfinite Recursion, Selected Topics in the Theory of Ordinals and Cardinals

Chapter 11 - Consistency and Independence in Set Theory