Russell's Attic

    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.

   (1995-03-29)