## 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.

Last updated: 1995-03-29

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**Russell's Attic**» Russell's Paradox » rusty iron » rusty memory