Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Matheology � 191
Posted:
Jan 13, 2013 5:44 PM
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In article
<0fa84faa-2103-4a2a-9c16-b6d498f23dd4@4g2000yqv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> > The definition of a set being countable is that there is a surjection
> > from |N to that set.
> >
> > Thus in order to PROVE a set is countable one must show a surjection
> > from |N to that set, which is just a listing, possibly with repetitions,
> > of that sets members.
> >
> > But any listing of the paths of a Complete Infinite Binary Tree (as
> > infinite binary sequences) proves itself incomplete.
> >
> > Thus the set of paths cannot be made to fit the "countable" definition.
>
> Above you see the enumeration of the set
>
> 0.
> 0.0
> 0.1
> 0.00
> 0.01
> 0.10
> 0.11
> ...
>
> Regards, WM
Since every path of a Complete Infinite Binary Tree must have infintely
many nodes (or, equivalently, infinitely many branchings), your list of
finite objects contains none of them
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