Zorn's lemma, the well-ordering principle and the axiom of choice are three equivalent propositions. It has been said that from appearances, the axiom of choice has to be true, the well-ordering principle has to be false and Zorn's lemma is too confusing to figure out. Zermelo Frankel formulated the axiom of choice in 1904 in an attempt to solve the first problem on Hilbert's famous list, the continuum hypothesis. The axiom states that given any collection of mutually disjoint sets there is a set that contains one element from each of the given sets. Zermelo proved that this axiom is equivalent to the well-ordering principle. His axiom was controversial at the time. Many hoped that it was unnecessary. Paul Cohen dashed these hopes when he proved (in 1963) that the axiom of choice is independent of Zermelo-Fraenkel set theory.

To explain these notions, a little set theory is...