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Re: Detecting whether an informal argument uses the axiom of choice
Posted:
Apr 12, 2012 6:00 AM
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On Thu, 12 Apr 2012, Paul wrote:
> I used to think that when a proof says something like "Choose any x
> such that x...." then the axiom of choice is being invoked, because x
> is not uniquely defined and we are asked to "choose" one non-
> constructively.
>
> However, this appears to be incorrect. Take the following argument
> that equipotence is an equivalence relation: "Let f be a bijection A-
> >B, and g be a bijection B->C then the composition of f and g is a
> bijection between A and C."
>
> Apparently, this does not require the axiom of choice even though it
> could be worded "There exists some bijection between A and B. Choose
> one of them and call it f".
>
It didn't state there's a bijection f:A -> B.
It stated if f:A -> B is a bijection.
So a bijection isn't being chosed, it's being assume.
> So the process of deciding whether an informal argument uses the axiom
> of choice is not clear to me.
>
For example, choosing an x for which x is in { 0,1,2 } doesn't require
the axiom of choice.
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