| **Definition:** | | An imaginary room containing countably many pairs of shoes (i.e. a pair for each natural number), and countably many pairs of socks. How many shoes are there? Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say). How many socks are there? Also countably many, we want to say, but we can't prove it without the axiom of choice, because in each pair, the socks are indistinguishable (there's no such thing as a left sock). Although for any single pair it is easy to select one, we cannot specify a general method for doing this. |