Most mathematicians, I suspect, regard the real numbers as, well, real — entities existing in the real world. They’re so familiar they’re even kind of boring. However, there are some very strange subsets of the real line — and the various Cantor sets are just the beginning. Beyond that, many questions about topological and measure theoretic properties of the real line are formally undecidable in the context of ZFC (Zermelo-Fraenkel plus the Axiom of Choice) set theory. When I think about this and about consequences of alternatives to the continuum hypothesis, I’m not at all sure about the reality of the real numbers. Being a real number agnostic is uncomfortable.
The book under review is addressed to experts and begins to address some of the questions about the complexities of the real line. It doesn’t — and probably can’t — address any of my visceral concerns, but it does dive deeply into some of the crucial questions. The author distinguishes four different — sometimes overlapping — periods in the history of investigations of the real numbers. In his interpretation, the first period extends from basic understanding of properties beginning as early as 2000 BCE and continues through Euler and Bolzano. The second period, concentrated in the second half of the nineteenth century, focuses on exact definition of the real number system and is driven by the beginnings of rigorous analysis. The third period involves an intensive study of the algebraic, topological and measure theoretic properties of the reals. Finally, the fourth period starts with an appreciation of the unanswered questions raised in the third period and focuses on set theoretical questions.
The author proposes to collect and present results from the second, third and fourth periods. His book first appeared first in 1979 in Slovak, and has only recently been published in English in a new and revised form as part of the Monographie Matematyczne series. The primary elements of the author’s investigations are set theory, topology, and measure theory. All of these are treated in a very sophisticated way. Analysts should be especially aware that there is a good deal of set theory here, and it is intensely in play throughout.
The first chapter summarizes Zermelo Fraenkel set theory and the main elements of point set topology that the author needs. A precise definition of the real line and a proof of its existence and uniqueness are provided in the next chapter. Chapters 3 through 8 focus on the classical theory of the structure of the real line, including topics like the Baire hierarchy, Lebesgue measure, the Borel hierarchy and analytic sets. Along the way there is an entire chapter on the apparent duality between measure and Baire category. The final chapter presents several questions about the structure of the real numbers that can be shown to be undecidable by forcing arguments. It is worth noting that the author is very cautious about using the Axiom of Choice; whenever possible, he uses the Axiom of Determinacy instead, an axiom that implies a weak form of the Axiom of Choice.
The author provides an appendix with background material on set theory, algebra, topology and metamathematics. He suggests in the introduction that readers can fill in their background using the appendix, read Chapter 1 and then skip around to follow their interests. This is a very optimistic view. Unless the reader is well-versed in non-trivial set theory (or willing to become so), this is a very challenging read.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.