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Topic: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR:
THE REALS ARE UNCOUNTABLE!

Replies: 47   Last Post: Jan 12, 2013 11:33 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted: Jan 10, 2013 4:35 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 10 Jan., 18:47, Zuhair <zaljo...@gmail.com> wrote:
> > On Jan 10, 1:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > So accordingly we
> > can discriminate a number of reals up to the number of all Omega_sized
> > tuples of finite initial segments of reals,

>
> As omega is not a finite number, you have no finite
> distingusihability.

Membership is a set, or a list, can be a finite problem even for
infinite sets or lists. And the nonpresence of Cantors antidiagonal in
any list from which it is constructed is a finite problem.
>
> > and this would be Aleph_0
> > ^ Aleph_0 and we have NO intuitive justification to say that the
> > number of such tuples is countable.

That depends on what definition of countability one uses.
For those using the standard definition, there is not only intuitive,
but formal, proof of countability of the set of members of a list and
formal proof of the non-countability of any set which demonstrably
cannot be listed.

>
> But there is a striking ground that is more fundamental than any wrong
> or correct logical conclusion, namely that you cannot find out any
> real number of the unit interval the path-representation of which is
> missing in my Binary Tree constructed from countable many paths. At
> least by nodes, you cannot distinguish further reals, can you?

We certainly cannot tell which ones are missing until WM tells us which
ones are present, which WM is careful never to do.

On the same basis, I can claim to have an infinite binary path (or a
real number) which is not in WM's tree (or his set of countably many
reals) and he cannot prove otherwise less I first tell him which
sequence (or real) I mean.

I have such a binary sequence, and I challenge WM to prove it is already
in his allegedly Complete Infinite Binary Tree. A mere claim of
completeness of his tree fails.
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