Any chance of a layman's explanation of the Axiom of Choice? I choose to ask this...
You have a drawer full of pairs of socks. Can you choose one sock from each pair? Yes: just take the pairs out one by one and choose a sock arbitrarily.
You have an infinite cupboard full of pairs of shoes. Can you choose one shoe from each pair? Yes: just choose the left shoe of each pair.
You have an infinite drawer full of pairs of socks. Can you choose one sock from each pair? Yes: just take the pairs out one by one and—
Wait—what does it mean to take the infinitely many pairs of socks out "one by one"? In what order? Might there even be too many socks to put them in
any order?
The axiom of choice says that you don't have to worry about the details: you have infinitely many choices to make, and you can just make them. There are lots of statements that are equivalent to the axiom of choice, though not always obviously. One says that every set can be put in a very nice order, one where at every stage (even if you've just taken infinitely many steps) there's guaranteed to be a well-defined "next" element in the order. (If you're wondering how this could possibly fail, imagine asking for the next integer after minus infinity, or the next real number after zero.)
People sometimes get hung up on consequences of the axiom of choice like the Banach–Tarski "paradox", which says that you can break two spheres into 5 "pieces" and reassemble them to get two spheres. But usually the problem is that you were thinking about something in the wrong way.
For example, the naive objection to Banach–Tarski is that you seem to be getting "more stuff" out of nowhere. But it turns out that Banach–Tarski is really just a cheat. You start by doing something essentially equivalent to saying that an infinite binary tree is basically two copies of itself glued together, then somehow embed this picture into 3D space. But the way the embedding is done means that all of the "pieces" are such a horribly mixed up collection of dust that we can't even sensibly talk about their size, so it's not clear that you have any "stuff" at all, never mind extra stuff.