I was learning Set Theory for fun and I came across something called the axiom of Choice,
What is the axiom of choice?
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3Have you tried googling it? – Michael Albanese Sep 17 '13 at 03:28
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Try reproducing the definition for starters. – Memming Sep 17 '13 at 03:28
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The axiom of choice says that there is a set of all left socks. If the two socks in each pair are indistinguishable, then there is no rule of deciding which sock in each pair is to be considered the left sock. The axiom of choice says that there is a set that contains exactly one member of each pair (and if there is one such set, then it follows that there are many).
One form of the axiom of choice says that in every set of pairwise disjoint non-empty sets, there is a set that contains exactly one element of each of them.
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7Incidentally (though this is moot now, of course) I know the analogy you're after but IMHO the opening phrasing muddles the point; if left and right socks are distinguishable then you don't need choice to build a set of all left socks, but if socks are indistinguishable then 'the set of left socks' doesn't really make sense as a concept. The mention of "a set that contains exactly one member of each pair" is much better phrasing. – Steven Stadnicki Sep 17 '13 at 04:03