  Zermelo’s original purpose in introducing AC was to establish a central principle of Cantor’s set theory, namely, that every set admits a well-ordering and so can also be assigned a cardinal number. Zermelo’s 1904 introduction of the axiom, as well as the use to which he put it, provoked considerable criticism from the mathematicians of the day.  The chief objection raised was to what some saw as its highly non-constructive, even idealist, character: while the axiom asserts the possibility of making a number  of—perhaps even an uncountable number—of arbitrary “choices”, it gives no indication whatsoever of how these latter are actually to be effected, of how, otherwise put, choice functions are to be defined. This was particularly objectionable to mathematicians of a “constructive” bent such as the so-called French Empiricists Baire, Borel and Lebesgue, for whom a mathematical object could be asserted to exist only if it can be defined in such a way as to characterize it uniquely. Zermelo’s response to his critics came in the form in two papers in 1908. In the first of these, as remarked above, he reformulated AC in terms of transversals; in the second (1908a) he made explicit the further assumptions needed to carry through his proof of the well-ordering theorem. These assumptions constituted the first explicit presentation of an axiom system for set theory.