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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 5:21 AM
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On 11 Jan., 10:39, Virgil <vir...@ligriv.com> wrote:
> In article
> <3810bc42-c275-4897-94ba-8280508e9...@10g2000yqk.googlegroups.com>,
>
> WM <mueck...@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.

Correct. But Cantor's list requires decimals or equivalent
representations.
>
> > 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.

I have not omitted any path that can be defined by a list of nodes.
>
> 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.

Since all finite paths are in the tree and since |N does not contain
an infinite n, there cannot be missing anything. If so, let me know
what is missing - from looking onto the covered nodes.
>
> But the set of all such representation is, by Cantor's diagonal proof,
> not countable, since countable = listable.

Cantor's diagonal proof proves that the set of distinguishable or
definable reals cannot be put in bijection with |N. There are
countable sets that cannot be in bijection with |N. Compare the set of
definable reals.
>
> > 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.

Call the finite initial segments finite paths. I construct the Binary
Tree by all finite initial segments.
>
> 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
> already endless by definition.

No one can find paths missing in the Complete Infinite Binary Tree
that contains all nodes covered by at least one path. Nevertheless
many claim to be able (but of course they are not).
>
> > 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.

Call the finite initial segments finite paths.

>
> And if Hilbert accepts infinity, WM will not overcome.

Hilbert said: das Unendliche findet sich nirgends realisiert; es ist
weder in der Natur vorhanden, noch als Grundlage in unserem
verstandesmäßigen Denken zulässig - eine bemerkenswerte Harmonie
zwischen Sein und Denken.

The infinite is nowhere realized; neither is it present in nature nor
admissible as the foundation of our intellectual/reasonable thinking -
a remarkable harmony between being and thinking.

You can learn it when you attend my lesson next week. I quote it
always with pleasure at the end of the semester as one of the joys of
life.

You see that the infinite cannot be the foundation of rational
thinking. So if it is used as a foundation (like the axiom of infinity
in ZFC), then the thinking based upon it cannot be called intellectual/
reasonable (according to Hilbert). Therefore I call it matheology.

Regards, WM

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