Prologue

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.