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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

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted: Jan 11, 2013 4:39 AM

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

> 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.

They can be, but they are quite often represented by other methods.

> 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.

We have no problem with all of your paths being in that tree, what we
have the problem with is the paths that you have necessarily omitted in
any construction that produces only countably many paths.

Every path in a Complete Infinite Binary Tree can be represented by an
infinite binary sequence, a mapping from |N to {0,1} and every such
mapping necessarily occurs in any representation of ALL paths.

But the set of all such representation is, by Cantor's diagonal proof,
not countable, since countable = listable.

> You will
> already understand this, when you know that I use every finite path.

There are NO finite paths in a Complete Infinite Binary Tree.

A path is, by definition, as long as possible already, so that no
extension of a path is possible

> But I append always an infinite extension.

No one can do it to a path, which in Complete Infinite Binary Tree is

> I don't tell you what this
> extension is in order to show you that the belief in its existence is
> simply nonsense.

It IS nonsense but only because your finite path is already 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".

In WMytheology WM may do as he likes, but in mathematics he does not
rule. And if Hilbert accepts infinity, WM will not overcome.
>
> Therefore there are not more than countably + countably = countably
> many paths in the Binary Tree.

While WM might, with better justification, oppose the very existence of
a Complete Infinite Binary Tree, he does not have the power to concede
its existence and then argue that it must not fit the very definition of
being a Complete Infinite 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

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