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Zermelo's Axiom of Choice: Its Origins, Development and Influence

Gregory H. Moore
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Joel Haack
, on

Zermelo’s Axiom of Choice is a Dover reprint of a classic by Gregory H. Moore first published in 1982. It provides a history of the controversy generated by Zermelo’s 1908 proposal of a version of the Axiom of Choice. Moore provides the philosophical and mathematical context for the controversy, carrying the story through Cohen’s proof that the Axiom of Choice is independent of the Zermelo-Fraenkel axioms for set theory. The story is told effectively and in great detail, giving this reader a real appreciation of the reasons for the controversy.

Moore makes it clear that the most significant concern among mathematicians regarding the Axiom of Choice stemmed from the expectation that mathematical objects should be explicitly constructed. Some mathematicians of the nineteenth century asked, “Which of the procedures valid for any finite number of steps … could be extended to an infinite number?” [p. 82] Moore’s analysis of this question includes four steps on the process toward abstraction: (1) selecting arbitrary elements from finitely many sets (dating back to Euclid); (2) making infinitely many choices via a stated rule; (3) making infinitely many choices via an unstated rule (for example, Cauchy in 1821 proving an Intermediate Value Theorem); and (4) making infinitely many choices where no rule was possible (for example, Cantor in 1871 showing the equivalence of continuity and sequential continuity).

The story that Moore provides proceeds both logically and chronologically. He begins with the prehistory of the Axiom of Choice, discussing those results that are either equivalent to the Axiom, that made an implicit use of the Axiom in a proof (whether necessarily or avoidably), or that motivated the introduction of axioms for set theory. The discussions in this portion of the book include mathematicians such as Gauss, Bolzano, Weierstrass, De Morgan, Peirce, Schröder, Borel, Jordan, Lebesgue, Hilbert, Bernstein, Peano, Hardy, Jourdain, and Zermelo. In short, it is a real who’s who of mathematicians of the period. Major results that are included in the discussion are the Well-Ordering Principle, Dedekind’s definition of infinite sets, the Trichotomy of Cardinals, Russell’s Paradox, Burali-Forti’s Paradox, and the (Generalized) Continuum Hypothesis. (These lists provide some sense of the breadth of Moore’s discussion and alert potential readers to topics that they might wish to pursue in this book.)

Zermelo explicitly stated the Axiom of Choice in 1904, introducing it as an assumption required for his proof that every set can be well-ordered. This result contradicted one supported by J. König, namely that the set of real numbers cannot be well-ordered. Thus Zermelo’s proof generated a great deal of controversy; a discussion of this is the content of Moore’s second chapter. Zermelo was led to publish an axiomatization of set theory, including the Axiom of Choice, to counter objections to his proof. Another consequence was the eventual formalization of mathematical logic and a discussion of the nature of the definitions of mathematical objects. In addition to many of the contemporary mathematicians listed in the previous paragraph, in this second chapter we hear of Hadamard, Hausdorff, Schoenflies, and Poincaré.

In the third chapter, Zermelo’s axioms for set theory appear explicitly. Rather than persuading the mathematical community of the acceptability of his proof that every set can be well-ordered, the proposed set of axioms themselves came under fire; the response was quite ambivalent. Moore discusses the disagreements during the period of 1908-1918 in detail. Additional mathematicians who appear here are Sierpiński, Edwin Wilson, D. König, E. Noether, Brouwer, Luzin, Suslin, Weyl, and Kuratowski. Among the results discussed are Hausdorff’s Paradox, Steinitz’s results concerning algebraically closed fields, Hartogs’ proof of the equivalence of the Trichotomy of Cardinals and the Axiom of Choice, and various maximal principles. In his summary section, Moore lays out four reasons why most mathematicians remained ignorant of the Axiom’s significance: 1) mathematicians didn’t yet understand how many of their arguments made implicit use of the Axiom; 2) it was uncertain whether use of the Axiom would lead to a contradiction; 3) there remained the controversy over what constituted the construction of a mathematical object; and 4) no one yet was an advocate, fully investigating the consequences of the Axiom.

In the fourth chapter, Moore provides the story of how the four reasons mentioned above were addressed in the period from 1918–1940. Here we learn of the Warsaw school’s investigations: Sierpiński and Tarski, among others, detailed how the Axiom was entwined with basic results in both set theory and real analysis. A wide-ranging list of results are discussed, including Tarski’s results on finite sets, results on cardinality and cardinal numbers, Zorn’s Lemma, the Banach-Tarski paradox and numerous applications within algebra, logic, set theory, and topology. A great variety of mathematicians appear in this chapter, including von Neumann, Lindenbaum, Chevalley, E. Artin, Bourbaki, Tukey, Teichmüller, Krull, Stone, Schreier, McCoy, G. Birkhoff, Hamel, van der Waerden, Fréchet, Lindelöf, Alexandroff, Urysohn, Tychonoff, Čech, E. H. Moore, Cartan, Tonelli, Cipolla, Levi, Viola, Church, Löwenheim, Skolem, Ramsey, Mirimanoff, Lindenbaum, and Mostowski. The chapter concludes with a discussion of both Gödel’s proof of the relative consistency of the Axiom of Choice with the Zermelo-Fraenkel axiomatization of set theory, and of this result’s reception.

The epilogue discusses first the continued investigations in set theory, algebra, graph theory, and logic that make use of the Axiom, then addresses Cohen’s proof of the independence of the Axiom of Choice from the other Zermelo-Fraenkel axioms. Moore concludes by quoting Dana Scott: “The Axiom of Choice is surely necessary, but if only there were some way to make it self-evident as well…  ” (page 310)

The first appendix contains the translation of five letters sent among Hadamard, Borel, Baire, and Lebesgue in 1905 that discuss the proof given by Zermelo in 1904 that every set can be well-ordered. The second appendix contains eleven tables displaying the deductive relationships among a variety of propositions related to the Axiom of Choice.

The Dover reprint of the 1982 Springer-Verlag edition is enhanced with a fifteen-page preface that sketches developments since the original release of the book. As Moore notes, a book published in 1998 listed the known relationships of 400 consequences of the Axiom of Choice, while there were more than 700 articles published between 1998 and 2013, when this Dover edition appeared. So it is not surprising that Moore is able to provide only a taste of more recent developments in the areas of topology, algebra, analysis, and set theory in this preface.

The book is admirable. Moore has provided a clear guide through the exciting controversies and developments related to the Axiom of Choice. The references to primary sources throughout the book buttress his conclusions.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.

List of Symbols
1. The Prehistory of the Axiom of Choice
2. Zermelo and His Critics (1904-1908)
3. Zermelo's Axiom and Axiomatization in Transition (1908-1918)
4. The Warsaw School, Widening Applications, Models of Set Theory (1918-1940)
Epilogue: After Gödel
Appendix 1. Five Letters on Set Theory
Appendix 2. Deductive Relations Concerning the Axiom of Choice
Journal Abbreviations Used in the Bibliography
Index of Numbered Propositions
General Index