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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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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!

Posted: Jan 11, 2013 1:51 AM

On 10 Jan., 22:35, Virgil <vir...@ligriv.com> wrote:

> > 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,

In mathematics reals are represented by sequences of digits or bits.
In the Binary Tree bits correspond to nodes. I can prove that every
path that can be defined by listing its nodes is covered by my
construction, i.e., it is contained in the Binary Tree. You will
already understand this, when you know that I use every finite path.
But I append always an infinite extension. I don't tell you what this
extension is in order to show you that the belief in its existence is
simply nonsense.

But even if you believe in infinite extensions and surmise that your
favourite extensions are not in my Binary Tree, then you should
understand, that there are only countably many infinite extensions,
because they require a finite definition like "always 0" or "bits of
the sequence of pi".

Therefore there are not more than countably + countably = countably
many paths in the Binary Tree.

> 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 would only ask for nodes. If all are there, then mathematics has no
tool to place further paths in the Binary Tree.
>
> 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.

I would not dare to guess what you have chosen because it is not
represented by nodes but only by a finite word of a countable set.

Regards, WM