had a cute
way of describing the Axiom of Choice. You need the axiom
to pick your socks
, but not your shoes
Allow me to elaborate. First of all, we must assume that we are talking about infinitely many pairs of socks and infinitely many pairs of shoes. The axiom is not necessary for finite collections of sets. Suppose then that we wish to form an infinite set of shoes containing a shoe from each pair of shoes. In this case we don't need the axiom because we can distinguish the individual shoes in each pair. This gives us an explicit method for picking our shoes. Our method, for example, could be "pick the left shoe from the first googolplex of pairs, and pick the right shoe from the the rest of the pairs." This doesn't require us to make any choice except the one we made at the beginning, before we began our shoe-picking procedure.
No such luck in picking our socks, however. Assuming our pairs of socks are pretty bland (and if we are mathematicians, they probably are), we have no way of distinguishing a left sock from a right sock. As we are going through our pairs of socks and picking a sock, we have to make infinitely many choices. Each time we get to a new pair of socks, we have to (somehow!) decide which one of the socks to put in our ongoing set. And this requires the Axiom of Choice.