Basic Set Theory

This book, a Dover reprint of a text first published by Springer in 1979, is a very rigorous and quite sophisticated introduction to axiomatic set theory. Given both the selection of some of the contents and the way in which this content is presented, this book’s title gives a whole new meaning to the term “basic.”

The book is divided into two parts. Part I (“Pure Set Theory”) covers an axiomatic introduction to Zermelo-Frankel set theory, both with and without the Axiom of Choice (denoted ZFC and ZF, respectively). Topics covered include cardinal and ordinal numbers and their arithmetic, as well as the Axiom of Choice and a number of its equivalents and alternatives. Part II (“Applications and Advanced Topics”) discusses ways in which set theory is used (for example, in topology) and also looks at more sophisticated topics in set theory. More about this shortly.

Some of the material in Part I is in some sense “basic,” but the way in which it is presented here is not. Although the back-cover blurb advertises this book, as back-cover blurbs inevitably do, as being “[g]eared toward upper-level undergraduate and graduate students,” I view this book as being wholly unsuitable as a text for an undergraduate course at an average university. The author’s writing style is succinct, and illustrative examples are few and far between. Learning the axioms of set theory from this text is a much more complicated matter than, say, seeing an axiomatic development of group theory or linear algebra. Some of this may be inherent in the nature of the subject matter, but much of it is attributable to the very sophisticated level at which the author approaches things.

For example: many books on axiomatic set theory begin with an introductory (“naïve”) account, to help motivate what follows. There is none of that here; the author simply begins with a discussion of “sets” versus “classes”. This all takes place very early in the text (the first ten pages or so), even before the axioms of ZF and ZFC have been set out.

In addition, the author uses the language of first order predicate logic with equality to discuss the axioms, and this sometimes results in very complicated-looking expressions that will undoubtedly be confusing to beginning students. The “schema of replacement” that appears on page 30, for example, which is much too cumbersome for me to reproduce here, takes one full line of text to write out. The use of first order logic to write the axioms of set theory is certainly not uncommon, but seems unsuitable for students with little background in logic: I can’t help but believe that such students would be overwhelmed by much of the discussion here.

Additionally, a considerable amount of the material in Part I of the book is certainly not what I would call “basic”. Indeed, there are, particularly in chapter V, some results that were, in 1979 when the book was originally published, only a few years old.

Part II of the text covers some advanced topics in set theory and also looks at ways in which set theory is applied to other areas of mathematics. The first chapter in this part is a very rapid (about 15 pages long) overview of point set topology, essentially devoid of proofs (except for one or two results where some brief hint of a proof is given). This chapter seems to be intended as background for some of the material in the next two chapters, the first of which discusses the real numbers and real spaces. Following an exceptionally succinct (four page long) description of the integers, rational numbers and real numbers, the author also introduces Cantor space and Baire space and discusses the topological and set-theoretical structure of them and the real numbers. Chapters like these two are not common in beginning set theory textbooks, but there may be some benefit in indicating how set theory impacts these topics.

The next chapter introduces Boolean algebras (“not a part of set theory proper, but … an off and on companion to set theory”), starting with the definition. Applications to topology are given (Tychonoff’s theorem is proved using filters and assuming the Axiom of Choice, and then later it is stated, with a proof sketched, that this theorem is actually equivalent to the Axiom of Choice), and related topics in set theory such as Martin’s axiom (a statement implied by the continuum hypothesis, but not itself provable in ZF, hence viewable as a weaker version of the continuum hypothesis) are introduced.

The final chapter in the text is on infinite combinatorics and large cardinals. An introduction to constructible sets is given, but the subject is not pursued in depth. People interested in a somewhat more accessible introduction to some of these ideas can also consult the last third of Combinatorics and Graph Theory by Harris, Hirst and Mossinghoff (which lists Levy in the bibliography as a “more technical” reference).

Though not, as I have said, suitable for undergraduates, the Levy book may fare better as a possible text for reasonably sophisticated graduate students (although it does deliberately omit some topics, such as model theory and forcing, that one might want to cover in such a course), and also provides a valuable reference for mathematics professionals in other specialties. In this regard, I was particularly impressed by two stylistic decisions made by the author. One was to include, after the statement of many definitions or theorems, a reference to the name of the person responsible, and the date of discovery or creation. (This is how I, a non-expert in set theory if ever there was one, was able to confidently assert, earlier in this review, that some of the results established here were proved just a few years before the text was originally published.)

Another nice feature is the inclusion of the phrase “Ac” before the statement of any theorem whose proof requires the Axiom of Choice. However, although the author spends some time discussing weaker versions of this Axiom, he does not distinguish between the full axiom and any weaker versions when annotating theorems in this fashion.

The book contains a fairly large number of exercises, scattered throughout the body of the text. There are no back-of-the-book solutions, but some of the more difficult ones have hints added to them.

I previously mentioned that the original Springer text was published in 1979. This Dover reprint was published in 2002. While the body of the text is basically unchanged, a six-page Appendix of additions and corrections has been added at the end, along with a one-page update to the original (quite extensive) bibliography.

To summarize and conclude: this is not a book for beginners to learn this material from for the first time, but people with some background and sophistication in the area should find much of value here. Levy’s expertise in this area is well-known, and he has obviously given a great deal of thought to how to present this material. This is certainly a book that belongs in any good college library.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.