As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. 



I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this, the Wikipedia entry is not in plain, simple, understandable language. Can someone give me a nice simple explanation of what this axiom is, and perhaps explain the XKCD joke as well?

The joke is really about the Banach-Tarski theorem, which says that you can cut up a sphere into a finite number of pieces which when reassembled give you two spheres of the same size as the original sphere. This theorem is extremely counterintuitive since we seem to be doubling volume without adding any material or stretching the material that we have.

The theorem makes use of the Axiom of Choice (AC), which says that if you have a collection of sets then there is a way to select one element from each set. It has been proved that AC cannot be derived from the rest of set theory but must be introduced as an additional axiom. Since AC can be used to derive counterintuitive results such as the Banach-Tarski theorem, some mathematicians are very careful to specify when their arguments depend on AC.

Here is a formal statement of AC. Suppose we have a set $W$ and a rule associating a nonempty set $S_w$ to each $w \in W$. Then AC says that there is a function 
$$f:W \to \bigcup_{w \in W} S_w$$
such that for all $w \in W$
$$f(w) \in S_w$$
