Set Theory for the Working Mathematician by Krzysztof Ciesielski, West Virginia University, London Mathematical Society Student Texts 39, Cambridge University Press, London, November 1997. 228 x 152 mm 240pp, Hardcover £37.50, ISBN: 0 521 59441 3, (also available in paperback ISBN: 0521594650).
For the moment I will give in to the temptation to compare the book under review with Hrbacek and Jech's Introduction to Set Theory (Monographs and textbooks in Pure and Applied Mathematics 85. Marcel Dekker 1984. MR 85m:04001, Zbl 555.03016), because its goals, as set out on the cover and in the preface, are similar and because it is one of the books that I have been recommending to people who want to know `what set theory is and what it can do for them'. For the rest of the review I will refer to this latter book as [HJ] and to the book under review as [C].
What the books have in common can be gleaned from the respective tables of contents; each covers the material one usually finds in an introductory set theory text: sets, relations, functions, (well-)orderings, cardinals, ordinals etc. In addition both books give an axiomatic development of set theory taking care, especially in the beginning, to deduce everything from the axioms.
It is more interesting to look at the differences.
[HJ] discusses how Cantor's concept of a set leads to Russell's paradox (`the set of sets that are not elements of themselves') and to imprecision (`the set of all the great twentieth century American novels'). The suggested way out of these problems is ``to formulate some of the relatively simple properties used by mathematicians as axioms, and then take care to check that all theorems follow logically from the axioms'' thus ``to end up with a body of truths about sets which includes,among other things, the basic properties of natural, rational and real numbers, functions, orderings etc., but as far as is known no contradictions.''Gödel's incompleteness theorems are mentioned in passing as a warning that the complete truth will never be known.
[C] quotes Frege's `universal comprehension axiom schema' and shows how it gives rise to Russell's paradox. The need for an axiomatic set theory thus made clear, the author states that the ZFC axioms are now considered the most natural and closes with full-blown statements of Gödel's incompleteness theorems, this time also to dash hopes of ever coming up with a provably consistent set theory.
[C] takes a utilitarian view and treats the Axiom of Choice as just another axiom, but mentions its non-constructive nature. [HJ] develops a large body of set theory without the Axiom of Choice. Chapter 9 of the book is devoted to this axiom, its well-known equivalents and their uses in mathematics.
Both books give proofs of the Hahn-Banach theorem and the theorem that every vector space has a base as well as a construction of a discontinuous additive function. [HJ] adds a discussion on the fact that results of this kind of result have led to the ``universal acceptance of the Axiom of Choice''.
[C] devotes two chapters to the author's own subject: set-theoretic analysis. The full power of the well-ordering theorem is brought to bear on the construction of weird subsets of and weird functions on the reals.
The final part deals with consistency and independence. In its first chapter the author sets the stage for Martin's Axiom and forcing by showing how many inductive constructions (of countable length) can also be done using the Rasiowa-Sikorski theorem: given countably many dense sets in a partially ordered set one can find a filter meeting all of them. For example, Cantor's theorem on the uniqueness of rationals as an ordered set is proved using the partially ordered set of finite order-preserving maps. We meet Martin's Axiom as a possible generalization of the Rasiowa-Sikorski theorem and we see some of its well-known consequences such as Suslin's hypothesis, the productivity of the ccc and additivity of measure and category. The diamond principle is introduced because it allows the construction of a Suslin line.
The final chapter then is devoted to forcing; we get proofs of the consistency of the negation of the Continuum Hypothesis, of the diamond principle and of Martin's Axiom.
[HJ] closes with two chapters of a completely different nature. The first of these delves into uncountable sets and proves a few combinatorial theorems such as Ramsey's theorem and the Pressing-down Lemma; it also deals with the measure problem and discusses measurable cardinals.
Without going into much detail the authors give us, in the final chapter, a few glimpses of further topics: the constructible universe, a short description of what forcing entails and a discussion of the interplay between large cardinals on one side and descriptive set theory and arithmetic on the other.
Set Theory for the Working Mathematician is not so well-written; it could have done with some rewriting and editing. In present days there is no need to show that the language of set theory can be used --- everybody uses it. Unfortunately not many mathematicians really need the results and techniques from set theory. I don't think that this book (nor any other) will convince a lot of people, who are not already using set theory, that set theory is important to their work.