Abstract
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.
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References
This and all other cited articles by Hilbert can be found in his Gesammelte Abhandlungen (Hilbert 1932–1935).
When the term “the Axiom” occurs hereafter, it always refers to the Axiom of Choice.
A set A can be well-ordered if there exists a relation S (where aSb may be read as “a is less than b”) which orders A and if each non-empty subset B of A contains an element b such that bSc for all other c in B. Such an element b is called the least element of B. The relation S orders A if for every a, b, c in A, (i) if aSb and bSc, then aSc, and (ii) aSb or a = b or bSa, and (iii) not aSa.
Cantor 1883a, 550. This and all other cited articles by Cantor can be found in his Gesammelte Abhandlungen (Cantor 1932).
Zermelo 1904, translated in van Heijenoort 1967.
Zermelo 1908 and 1908a; both are translated in van Heijenoort 1967.
An urelement (also called an individual or an atom) was an object which contained no elements, belonged to some set, and yet was not identical with the empty set.
See Appendix 2 as well as Rubin and Rubin 1963.
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© 1982 Springer-Verlag New York Inc.
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Moore, G.H. (1982). Prologue. In: Zermelo’s Axiom of Choice. Studies in the History of Mathematics and Physical Sciences, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9478-5_1
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