100 years of Zermelo's axiom of choice: what was the problem with it?

Cantor conceived set theory in a sequence of six papers published in the Mathematische Annalen during the five year period 1879-1884. In the fifth of these papers, published in 1883, 1 he stated as a law of thought (Denkgesetz) that every set can be well-ordered or, more precisely, that it is always possible to bring any well-defined set into the form of a well-ordered set. Now to call it a law of thought was implicitly to claim self-evidence for it, but he must have given up that claim at some point, because in the 1890's he made an unsuccessful attempt at demonstrating the well-ordering principle. 2 The first to succeed in doing so was Zermelo, 3 although, as a prerequisite of the demonstration, he had to introduce a new principle, which came to be called the principle of choice (Prinzip der Auswahl) respectively the axiom of choice (Axiom der Auswahl) in his two papers from 1908. 4,5 His first paper on the subject, published in 1904, consists of merely three pages, excerpted by Hilbert from a letter which he had received from Zermelo. The letter is dated 24 September 1904, and the excerpt begins by saying that the demonstration came out of discussions with Erhard Schmidt during the preceding week, which means that we can safely date the appearance of the axiom of choice and the demonstration of the well-ordering theorem to September 1904. Brief as it was, Zermelo's paper gave rise to what is presumably the most lively discussion among mathematicians on the validity, or acceptability , of a mathematical axiom that has ever taken place. Within a couple of