Russell's Attic

mathematics 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)