Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice? I mean the following quote:
To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here).
Finite choice without AC
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set-theory
axiom-of-choice
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3Do you know how to choose one sock from one pair? (Presumably you do this every morning.) – 2011-09-13
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0In addition to my answer here, you might find some more information in the [possibly-too-complicated answer](http://math.stackexchange.com/questions/52587/what-are-the-axiom-of-choice-and-axiom-of-determinacy/52591#52591) I wrote here. In particular, in the "Added" part toward its end. – 2011-09-13
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0@Qiaochu: Why do you ask? Is the answer to the question as simple as what I do every morning? :) – 2011-09-14
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0As far as I understand I should accept one of the answer, but I don't know how to decide which one is better, I am not yet well acquainted with the topic... – 2011-09-14
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0@cabbage: yes, more or less. I suppose it depends on how formal you want an answer to be. – 2011-09-14