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Three Books on Set Theory:

Basic Set Theory, by A. Shen and N. K. Verschagin	Sets for Mathematics, by F. W. Lawvere and R. Rosebrugh	Basic Set Theory, by A. Levy

Reviewed by Michael Berg

Roughly, Shen and Verschagin presuppose the reader to be a future mainstream mathematician on the verge of hard-core upper division work, serious about the subject, but not necessarily a fledgling set-theorist. Their lovely little book (all of 116 pages) does a truly marvelous job in covering what every one in the game should know, whether he be an analyst, geometer, algebraist or number theorist — or anything else, for that matter. It's all there, from Cantor's theory of cardinals to transfinite induction, from Zermelo to Zorn.

But allow me to get down to a brass tack or two. The Department of Mathematics and Mechanics at Moscow State University is of course legendary. A colleague, who was once a dissident in the former USSR and came to the US as a refugee, said to me not long ago that it was the best place in the world for mathematics, once upon a time (before the fall of the Soviet Union). Perhaps he is right. Doubtless it must have been a contender. And the book under review "is based on notes from several undergraduate courses the authors offered for a number of years" at this marvellous department. Therefore it is a terrific book and does everything right: its selection of topics is not only logical, it is elegant, and the coverage is superb. But it is not for the timid. The problems are very nice: interesting and non-trivial (to put it mildly) and they supplement the main body of the text very well. But occasionally the problems are, well, spicy. However, reading this book while doing no problems is not only an act of intellectual violence, it is at odds with the authors' construction of the book in that the problems are strategically woven through the text and to omit them would introduce jump discontinuities. As a connected whole, the book is a pedagogical marvel.

So the book would be perfect for self-study, if the (young) student is very, very smart and already in love with mathematics. As regards class-room use, it would also be a marvellous experience (one I covet, actually) to use the book in a first course on set theory, say, at the sophomore or even junior level. But, again, these should be strong students. They'd have a great time, though: which other book on set theory (at this level) has the temerity to prove that a cube and a regular tetrahedron of the same volume fail to be equidecomposable? After an apt discussion of Hamel bases, this famous theorem arises as a pretty exploration of the notion of volume (and the authors go on to show a forteriori how Hamel bases can be avoided). A very nice bit of work.

By the way, I very recently used the book's proof of the existence of a Hamel basis for any vector space in my course on Advanced Linear Algebra. It is an extremely slick and quick argument, avoiding Zorn's Lemma in favor of an "in your face" invocation of the Axiom of Choice (less jarring to the novice, I think). And the discussion given in the book is typical of the entire book: to the point, elegant, and complete.

Obviously I highly recommend this book. As I said, it covers the basic set-theoretic tool-kit every mathematician should carry around at all times, and does so with style. And then there are all the beautiful applications, challenging and elegant problems, and even a lot of surprises — consider, for example, the very last problem in the book: "Find an error in the following 'proof' [of] the negation of the Continuum Hypothesis." What more can one ask for?

Now we come to Lawvere and Rosebrugh's idiosyncratic book. Evolving out of SUNY lecture notes in the 1980's the book apparently came into being at the urging of Saunders MacLane and therefore it is no surprise (also given some of Lawvere's other recent work) that it is heavily category theoretic in flavor. Indeed, the authors say in the Foreword: "This book is for students who are beginning the study of advanced mathematical subjects such as algebra, geometry, analysis, or combinatorics. A useful foundation for these subjects will be achieved by openly bringing out ... what they have in common." And they go on to say that "the resulting idea of categories of sets is the main content of this book."

As such, this book, very well (and clearly) written and of reasonable length (at 261 pages), should appeal to that proper subset of mathematicians who champion category theory but the book will probably be eschewed by the complementary set as fundamentally misguided. I belong to the former set in that I do adore category theory and believe it to be, among many other good things, a wonderful means for becoming able to see deep structural similarities within mathematics, ignoring even natural borders in sometimes dramatic ways. Beyond this there is great and austere beauty to be had in the structures themselves, as dealt with in category theory proper, even if it all begins with "first... deplet[ing] the object of nearly all content," as Lawvere and Rosebrugh put it (perhaps somewhat perversely).

This having been said, my claim is that a via media is in effect in that this book ought to serve quite well as a major player in the early education of a mathematician who sees, say, algebraic geometry in his future, or certain obvious parts of algebra or topology: the list is long (and getting longer). But such a student would do very well to go though a book like Shen and Verschagin's first.

Finally we come to Levy's book, carrying the same title as Shen and Verschagin's. It just goes to show that the word "basic" is not well defined. For Levy's meaning is narrower and his focus is very different: "Almost all ... books on set theory are of one of the following two kinds. Books of the first kind treat set theory on ... [the] level needed for studying point set topology and Steinitz' theorem on ... algebraic closure[s]. Books of the second kind ... give a more or less detailed exposition of several areas of set theory that are subject to intensive current research." While he does not overtly place it in either class, Levy says that his Basic Set Theory is "a book on a rather advanced level covering the basic material in an unhurried pace." So it deals with material basic to a set theorist properly so-called and thus differs profoundly from the other two books in this review, both in intent and scope. The audience in this case is really very narrow: fledgling set theorists and their fellow travellers. In that context it is a terrific book! It is very thorough and complete (and meaty, at 398 pages) and deals with a lot of very, very good stuff. Its second half, titled "Applications and Advanced Topics," covers such arcana as Baire category, Boolean algebras, and infinite combinatorics. A separate section on "The close relationship between the real numbers, the Cantor space, and the Baire space" indicates the intended reach of this work: it aims (for example) at getting at some of the more profound subtleties in the subject such as the notion of measure on the real numbers.

So it is, then, that Levy's book is also very different in style from the books discussed above: it is written by a set-theorist for future set-theorists (or, as I already indicated, their fellow-travellers — of which I was one once).

In closing, let me note that all three of these books are, in their own way, wonderful but the intersection of their intended (or proper) audiences, while non-empty (I know of at least one element), is very small, both in measure and in category. I recommend them all enthusiastically and hope to be able to use a non-empty sub-set of them in my own teaching.

Publication Data:

Basic Set Theory, by A. Shen and N. K. Verschagin. Student Mathematical Library, volume 17. American Mathematical Society, 2002. Paperback, 116 pp. $21 ($17 to AMS members). ISBN 0-8128-2731-6.

Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2002. Paperback, 250 pp., $35, ISBN 0521010608. Hardcover, 250 pp., $100, ISBN 0521804442.

Basic Set Theory, by Azriel Levy. Dover, 2002. Paperback, 398 pp., $21.95. ISBN 0486420795.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Posted July 25, 2003

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