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Re: Mathematics in brief
Posted:
Dec 9, 2012 2:39 PM
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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
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