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Re: Detecting whether an informal argument uses the axiom of choice
Posted:
Apr 13, 2012 10:53 PM
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"Richard Tobin" <richard@cogsci.ed.ac.uk> wrote in message
news:jm9hja$11tt$1@matchbox.inf.ed.ac.uk...
> In article <jm91mj$em6$1@news.albasani.net>,
> Peter Webb <r.peter.webbbbb@gmail.com> wrote:
>
>>> For selecting an element from a single non-empty set, we don't need
>>> the axiom of choice because the existence of an element of a non-empty
>>> set is true by definition.
>
>>Its existence is known, but that doesn't mean there is (to use your words
>>above) an "obvious function to select an element". There is no way of
>>producing an explicit function which always picks a Real from an arbitrary
>>non-empty subset of Reals.
>
> I am somewhat unsure about this. Does the function need to be
> explicit, or do we just need to show that one exists? I find several
> pages on the web asserting that this argument is sufficient:
>
> Let S be a non-empty set
> Then there exists an element x of S
> So the function that maps S to x is a choice function.
>
> For example:
>
> http://en.wikipedia.org/wiki/Axiom_of_choice#Restriction_to_finite_sets
> https://nrich.maths.org/discus/messages/114352/115218.html?1169570756
>
> http://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf
>
> If that argument is correct, the question then is why a similar
> argument doesn't work for all collections of sets:
>
> Each Sn in C is a non-empty set
> Then for each Sn there exists an element xn of Sn
> So the function that maps each Sn to xn is a choice function.
>
> One of the pages above says that the problem that the first line
> really needs to be an infinite list of assumptions, but why?
>
> -- Richard
I don't know.
But if I am thinking of an arbitrary, infinite subset of R, there is no
function that will explicitly choose a single element of this set. Unlike a
an arbitrary subset of N, where such a choice function does exist - pick the
smallest element.
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