Date: Dec 9, 2012 2:39 PM Author: Zaljohar@gmail.com Subject: Re: Mathematics in brief On Dec 9, 10:37 pm, Zuhair <zaljo...@gmail.com> wrote:

> On Dec 9, 9:45 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

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> > On 9 Dez., 18:40, Zuhair <zaljo...@gmail.com> wrote:

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> > > On Dec 9, 1:14 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

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> > > > On 9 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote:

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> > > > > On Dec 9, 11:35 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 9 Dez., 05:55, Zuhair <zaljo...@gmail.com> wrote:

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> > > > > > > You need to prove that the set of all paths is countable, and so far

> > > > > > > you didn't present a proof of that.

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> > > > > > The set of all finite paths is countable. Therefore it is not possible

> > > > > > to define an infinite path by adding nodes to any finite path. All

> > > > > > nodes to be added are already in finite paths. Therefore, by following

> > > > > > the nodes of a path, you never define an infinite path. It is

> > > > > > interesting that practically everybody not yet brainwashed can

> > > > > > understand that.

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> > > > > Every node is reachable by a finite path, that's correct. But that is

> > > > > irrelevant

> > > > > here, we are speaking here about the number of all "path"s in the

> > > > > Binary Tree

> > > > > and not about the number of all nodes. we know that the number of all

> > > > > nodes

> > > > > is countable, the question is: is the number of all paths (finite and

> > > > > infinite)

> > > > > is countable?

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> > > > So it has become obvious now, that is not possible to define "all

> > > > paths" by nodes. Only the finite paths can be defined by their nodes.

> > > > How can you define all paths if not by nodes?

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> > > Simply there are non finitely definable paths.

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> > No they are not anywhere. Your assertion is simpky false. I construct

> > one path through each node such that every node has its own path. (It

> > is irrelevant, which and how many other nodes belong to that path.) By

> > this construction every node is covered by its own path. And there is

> > no chance to define any further path by further nodes. There are no

> > further nodes available.

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> > > Anyhow what is the proof that ALL reals can be represented by paths of

> > > an infinite Binary Tree (actually two trees). It looks that only a

> > > countable subset of reals can be represented in that way. I'm not sure

> > > really.

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> > Only a countable subset can be represented by the Binary Tree. The

> > reason is that no path is really actually infinite.

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> Then you are not addressing what Cantor was speaking about, he is

> speaking about reals represented by ACTUALLY infinite sequences (paths

> in your case). It is clear that the set of all reals represented by

> FINITE sequences is countable, but those are just a very small subset

> of the set of all reals.

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> If one assumes Actual infinity, then it is easy to recover the

> diagonal path from any bijection between the reals and the set of all

> paths of the infinite binary tree, and this will be a path that is not

> present in the tree of course.

Sorry I meant between the naturals and the .....

You will need uncountably many infinite

> binary trees to recover all the reals. And again you simply failed to

> demonstrate a clear contradiction with Cantor's argument.

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> What you are not getting is that uncountability of the reals is a

> PROVED issue, it is proved in very weak fragments of second order

> arithmetic that are PROVED to be consistent. I don't know if you

> really get what I'm saying here.

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> However on the other hand still you can get countable models of those

> theories where the set of all reals can be defined. So both

> countability of the reals and uncountability of reals are open

> possibilities and can be spoken about by consistent discourses. So

> both are pieces of mathematics.

> Everything depends on the model you are working in.

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> > > Anyway the diagonal argument of Cantor is provable in very weak

> > > systems of ZFC which are proved to be consistent. So uncountability of

> > > reals is a possibility.

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> > And it leads to a contradiction with the fact that all real numbers

> > that are paths in the Binary Tree form a countable set.

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> > > Of course also Countability of reals is a possibility! since we can

> > > have countable models of ZFC or any theory that can define all the

> > > reals.

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> > I talk about *the* real numbers, which Cantor proved uncountable.

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> > > > It is a shame that someone defends the concept of "undefinable

> > > > number", unthinkable thought, anusable use, - and nevertheless claims

> > > > to be a logican and mathematician!

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> > > This includes the main bulk of experts on foundations of mathematics.

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> > The main bulk of experts on foundations of astrology is by far more

> > trustworthy.

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> > > Actually I see this statement of yours really unsubstantiated.

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> > You believe in undefinable numbers. But what should that belief be

> > good for???

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> They are good for letting you know that you cannot place the reals

> with the naturals in one-one correspondence in some models.

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> We can believe anything we like of undefinable numbers

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> > like of unicorns. Cantor's proof concerns definable numbers only! So

> > undefinable numbers do not support your standpoint anyhow.

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> > Regards, WM