Hilbert calls the Axiom of Choice the "most attacked up to the present in mathematical literature," Fraenkel characterizes it as "the most discussed axiom of mathematics second only to Euclid's axiom of parallels," and Kanamori sees it as the beginning of "abstract set theory." I'll start will a brief historical overview and then trace Zermelo's 1904 proof of the Well-Ordering Theorem
If PowerSet(X) has a choice function, then X can be well ordered.
using the Axiom of Choice.
As time permits, I'll talk about important consistency results and offer examples of implicit use of AC before Zermelo's 1904 formulation. Also, I have proof sketches of various equivalences (e.g., the well-ordering theorem and the multiplicative axiom, AC and the theorem that any set has a denumberable subset). Other topics include set-theoretic implications of AC and the link between AC and von Neumann's theory of ordinals. I'll conclude with some points on the history and controversy surrounding AC.