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What "choice" does the Axiom of Choice permit us to make? I've searched high and low and not found a satisfactory answer. To me it seems to add no new information to its hypotheses: Given an arbitrary collection of non-empty sets, isn't it true, without assuming anything and just by the fact that each set has at least one element, that you can map each set to an element within itself? Or is the rub that the statement is not provable or disprovable from the axioms of ZF (which I wish I were more familiar with)?

One author suggests that you need a hard-and-fast rule (e.g., if I'm using the domain of positive-length real intevals let f(interval) = the midpoint), but then says that we don't need AC for any finite interval because we are able, by induction, to pick set-by-set within the domain. I guess I need a more precise definition ...

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# Confusion over Axiom of Choice.

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