Zermelo’s Axiom of Choice

David Hilbert once wrote that Zermelo’s Axiom of Choice was the axiom “most attacked up to the present in the mathematical literature...” [1926, 178].¹ To this, Abraham Fraenkel later added that “the axiom of choice is probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago” [Fraenkel and Bar-Hillel 1958, 56–57]. Rarely have the practitioners of mathematics, a discipline known for the certainty of its conclusions, differed so vehemently over one of its central premises as they have done over the Axiom of Choice. Yet without the Axiom, mathematics today would be quite different.² The very nature of modern mathematics would be altered and, if the Axiom’s most severe constructivist critics prevailed, mathematics would be reduced to a collection of algorithms. Indeed, the Axiom epitomizes the fundamental changes—mathematical, philosophical, and psychological—that took place when mathematicians seriously began to study infinite collections of sets.

Throughout its historical development, mathematics has oscillated between studying its assumptions and studying the objects about which those assumptions were made. After the introduction of new mathematical objects, it often happened that the assumptions underlying them remained unspecified for a considerable time; only through extensive use did such assumptions become sufficiently clear to receive an explicit formulation. Usually a body of theorems, consequences of an assumption, were obtained before the assumption itself came to be recognized. At times, indeed, an assumption was specified precisely in order to secure a particular theorem or theorems. Of course, such an assumption ordinarily formed part of a nexus of suppositions with varying degrees of explicitness. What, one may ask, has caused such an assumption to become conscious and explicit? The question grows more complex as soon as we recognize that there was rarely, if ever, a single way of expressing an assumption and that various weakenings or strengthenings of an assumption could serve different mathematical purposes. This preliminary chapter explores how the use of arbitrary choices led, over most of a century, to Zermelo’s explicit formulation of the Axiom of Choice.

When in 1904 Zermelo published his proof that every set can be well-ordered, many questions lay unresolved. In the wake of Russell’s paradox, published in 1903, it was even uncertain what constituted a set. Moreover, Zermelo’s proof itself raised a number of methodological questions: Was it legitimate to define a set A in terms of a totality of which A was a member, as Zermelo had done? Did the class W of all ordinals invalidate Zermelo’s proof and entangle it in Burali-Forti’s paradox? Most important of all, was his Axiom of Choice true? Was it a law of logic? Should one postulate simultaneous, independent arbitrary choices in preference to successive, dependent ones? Did the cardinality of the set of choices affect the validity of the Axiom, so that the Denumerable Axiom was true but not the Axiom of Choice in general?

As early as 1896, even before the discovery of set-theoretic paradoxes, a few mathematicians had suggested that set theory ought to be axiomatized. Yet interest in formulating such an axiomatization remained very faint even in 1903, when Russell restated Burali-Forti’s result of 1897 as a paradox, and published his own paradox as well. Hilbert, for example, viewed Russell’s paradox as revealing that contemporary logic failed to meet the demands of set theory.¹ Russell asserted further that a solution to the paradoxes would result only from a reappraisal of the assumptions used in logic, rather than from technical mathematics [1906, 37]. Unperturbed by the paradoxes, Zermelo concentrated on axiomatizing set theory within mathematics rather than on revising the underlying logical assumptions.

When, shortly after 1918, a school of mathematicians emerged at Warsaw under Sierpiński’s tutelage, the Axiom finally gained the attention that it deserved, and thereby became the object of much careful research. Within a few years Polish mathematicians discovered interconnections between the Axiom and many other propositions in various branches of mathematics. Sierpiński’s survey of the Axiom’s uses was soon followed by Tarski’s research on definitions of finite set whose equivalence required the Axiom. Furthermore, Tarski discovered that each of several propositions in cardinal arithmetic implies the Axiom and hence is equivalent to it. On the other hand, Banach and Tarski extended Hausdorff’s paradox by demonstrating via the Axiom that any sphere S can be decomposed into a finite number of pieces and reassembled into two spheres with the same radius as S. Neither author regarded this result as any slight on the Axiom, but later mathematicians were to call it the Banach-Tarski paradox.

The history of the Axiom of Choice is the history of how an assumption’s status can change. At a given point in time, each assumption in mathematics forms part of a nexus of suppositions with varying degrees of explicitness. Throughout history, mathematicians have operated within conceptual frameworks in which certain assumptions were stated but in which, on the other hand, certain assumptions were tacit or even unconscious. Euclid, among others, presupposed properties of the continuum that were not recognized explicitly as necessary for geometry until the nineteenth century. During the late nineteenth century, Frege emphasized the need to make all one’s assumptions explicit, and Hubert’s axiomatization of geometry furthered this process by using a formal axiomatic method as distinct from that of Euclid. Zermelo’s Axiom of Choice is best viewed as a further attempt to formulate an implicit assumption explicitly.