Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Matheology � 191
Posted:
Jan 13, 2013 3:56 PM
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In article
<3c651dfd-5d3d-4464-bb03-fea1c590207b@10g2000yqk.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 13 Jan., 13:15, Zuhair <zaljo...@gmail.com> wrote:
>
> > What mean nothing more than saying that we have Countably many FINITE
> > paths
>
> Yes, and it is not intuitive nor needs it any formalization to
> recognize that everything that happens in Cantor-lists happens withing
> finite paths (or sequences of digits). It is absoluteley impossible
> that something happens elsewhere! And if a list contains all possible
> finite paths (which is possible as they are countable) then Cantor's
> "proof" proves the uncountability of a countable set.
Not unless the anti-diagonal is finite, which it can't be, since those
finite "paths" being listed do not, and can not, have a finite upper
bound on their lengths, so there cannot be an finite upper bound on the
anti-diagonal's length.
The obvious flaw in WM's argument shows how poorly he grasps what is
really going on.
>
> Note again: everything that happens in a Cantor-list happens withing
> finite paths or finite initial segments of the anti-diagonal.
But there is still no finite upper bound on the segments of the
anti-diagonal required. Thus even with all finite entries to the list
its anti-diagonal must be infinite.
>
> And please do me a favour and stop parroting of uncountable sets
> unless you can explain how something can happen *after* all finite
> initial segments.
I will continue to speak of uncountable sets until you can show that the
set of all infinite binary sequences ( or functions from |N to {01}) can
be listed with no anti-diagonal missing.
And as you cannot do this, you lose!
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