All undergraduate mathematics majors are exposed to the basic language of set theory, usually in their “transition to abstract mathematics” course. Either in that same course or early in a course on analysis, they also learn about the distinction between countable and uncountable sets. And, even if only in hallway conversations, they often learn about Russell’s paradoxical “set of all sets that are not members of themselves.” The latter usually appears only as a cautionary tale: doing set theory correctly requires some caution that leads to “axiomatic set theory.” And there, for most undergraduates, the story ends.

Every so often, however, an ambitious undergraduate or group of undergraduates wants to learn more. This book is intended to meet that need. It provides a careful introduction to axiomatic set theory that is accessible to (smart and well-motivated) undergraduates.

Most graduate textbooks in axiomatic set theory require a large amount of mathematical logic just to get started. The one I tried to read as an undergraduate, Takeuti and Zaring’s *Introduction to Axiomatic Set Theory* (volume 1 in the *Graduate Texts in Mathematics* series from Springer) is a good example: the first chapter is called “Language and Logic” and the only reason I could get through it is that I had taken a course in mathematical logic, something that few of our majors do. The book also introduces a lot of its own notation and presents many of its proofs in pure symbolic form. (Look, ma, no words!) I think I managed to get as far as the chapter on cardinals, but then I got lost.

Hrbacek and Jech successfully avoid most of that, relying only on a couple of pages where they provide an intuitive explanation of what is meant by “a property” and talk about quantifiers and logical connectives. The idea is that mathematics students already have a good sense of all that, and we can reserve the more delicate issues for later. This makes it easy for students to get started.

The presentation is gradual and well thought out. One of the principles Hrbacek and Jech clearly had in mind is to do as much as possible with the available axioms, introducing new ones only when the motivation for them is available. This is revealing. The first chapter sets up the really fundamental stuff: Existence, Extensionality, the schema of Comprehension, Pairs, Unions, and Power Sets. Note that the authors do discuss the difference between an axiom and an axiom schema, but do it lightly. Nothing in this chapter will surprise any of our students except the care with which things are presented. These axioms are enough to introduce relations, functions, and orders, which is what the authors do next, but not enough to guarantee the existence of any infinite set, so the third chapter adds the Axiom of Infinity.

It continues like that. Ordinals are introduced up until the point where one can’t proceed without the axiom schema of Replacement, which is then introduced to finish off the account. Cardinality forces the addition of the Axiom of Choice, which gets a careful discussion in chapter eight. One of the surprises for me was that the last axiom to be introduced is the axiom of Foundation, on the very good grounds that it isn’t really needed. (To be precise: all of ZFC can be read as the theory of the well-founded sets, and doesn't seem to be affected by allowing non-well-founded sets to exist.)

The authors emphasize the links between set theory and the rest of mathematics. There are chapters dedicated to the finite/countable/uncountable distinction, to the construction of the integers, rationals, and reals, and to the examination of the topology of subsets of \(\mathbb{R}\). There are more advanced chapters on filters and ultrafilters and on combinatorial set theory, highlighting other areas where set theory touches on other kinds of mathematics. Of course, these are chosen because they connect nicely with the model theory issues that will be mentioned in the last few chapters.

Things get fairly hairy towards the end, when the authors offer hints of several directions in which the theory might proceed: large cardinals, infinite combinatorics, descriptive set theory, infinite games, non-well-founded sets, consistency and independence results. This is more difficult and less familiar than what came before. At times I found it hard to care. For example, if all of standard mathematics can be done without inaccessible cardinals, does it really matter whether they exist? Why should I want the theorem that says that all uncountable Borel sets contain a perfect subset to extend to all projective sets? Does it really matter that some \(\Pi_2^1\) sets are not Lebesgue measurable? A lot of the authors’ arguments seem to stem from the assumption that there should be as many sets as possible and that they should behave as well as possible. Could be, but I’m not convinced. Nevertheless, the very fact that such questions pose themselves is proof that one learns a lot by reading the book.

If I had the chance to teach a course from this book, I would aim to cover at least the first ten chapters, probably moving fairly quickly through the “known” material in chapters 2, 4, and the early parts of 10. Then I’d probably jump to 14 to introduce the Axiom of Foundation and, if there was time, talk a little bit about consistency and independence. It would be fun to do.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME and is the editor of MAA Reviews.