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Replies: 180   Last Post: Jul 14, 2011 10:19 PM

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 Apollo Posts: 34 Registered: 6/26/11
Posted: Jun 26, 2011 4:42 AM

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

> On 25 Jun., 22:23, Uergil <Uer...@uer.net> wrote:
> > In article
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > On 25 Jun., 19:45, Rotwang <sg...@hotmail.co.uk> wrote:
> >
> > > > I = {P in P(T) | P is a path and P is infinite}
> >
> > > > is uncountable.-
> >
> > > Why do you think it is uncountable?
> >
> > Because it has been proved not to be countable, where countable means
> > capable of being bijected with the set of natural numbers.

>
> The atoms of Andromeda galaxy are not countable because you cannot
> biject them with the naturals. Nevertheless they form a set that is
> less than uncountable.

There is a difference between what is practically impossible and what is
in principle impossible. And if one could freeze time, so that the
number of atoms in the Andromeda galaxy remained constant, then they
could, at least in principle, be finitely counted.
> >
> >
> >

> > > You think so, because in Cantor's list, the anti-Diagonal D is not in
> > > the lines L_1 to L_n for every natural number n, and you conclude from
> > > that finite initial segment, that D is not in the complete list.

> >
> > Is there any element of that complete list that is NOT in some finite
> > initial sublist? Unless there is, and you can name it,

>
> That is your error. Wjy should it be possible to name it?

It is the impossibility of naming what does not exist which is the issue.

Since the question raised is whether such an element exists, you must
either name it or otherwise prove its existence before asserting its
existence.

In fact no such element can possibly exist. Each element has a position
which can be numbered by a natural so is in the finite sublist of that
number of elements.

So that any arguments that rely on its existence fail.

> Just the opposite is true. By induction we can prove that for every
> named natural number there are infinitely many natural numbers without
> names.

Irrelevant even if true.
>
> >
> > > But we can set up the same argument with the Binary Tree.
> >
> > And it shows, in much the same way,  that the set of paths in the
> > complete infinite binary tree is not countable.

>
> My argument shows that there is no infinite path in any initial
> segment B_k.

That is irrelevant, as every path contains infinitely many nodes not in
any B_k.

> Further it shows that every node is in some initial segment B_k.

But not any paths.

> Further we know that an infinite path has infinitely many more nodes
> than every finite path.
>
> This contradicts the assumption that either all nodes are in finite
> initial segments or that actually infinite paths do exist at all.

No it doesn't.

Every natural is in a finite initial segment of naturals, but an
infinite set of naturals exists (example of a unary tree)

>
> Same is true for Cantor's list.
>

> > By your argument, one would equally have to conclude that the set of all
> > naturals is finite.

>
> Of course there is no completed infinite set.

In some set theories, they are provided by the axioms and therefore MUST
exist.

> The set of natural
> numbers is unfinished, i.e., potentially infinite.

Not in proper set theories.

In the sort of chaotic ones that inhabit WM's mental universe, all
sorts of ugly things seem to occur and much that is beautiful in
mathematics does not appear.

the rest of WM's garbage conclusions snipped!

Date Subject Author
6/24/11 Newberry
6/24/11 Brian Q. Hutchings
6/25/11 Virgil
6/25/11 Newberry
6/25/11 Virgil
6/25/11 mueckenh@rz.fh-augsburg.de
7/1/11 LauLuna
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7/2/11 mueckenh@rz.fh-augsburg.de
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7/2/11 Virgil
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7/2/11 Virgil
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