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Re: Mathematics in brief
Posted:
Dec 10, 2012 4:17 AM
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On 10 Dez., 09:07, Virgil <vir...@ligriv.com> wrote:
> In article
> <5a2d9b2e-c558-446a-908f-1a5f24d3f...@r14g2000vbd.googlegroups.com>,
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 10 Dez., 06:32, Zuhair <zaljo...@gmail.com> wrote:
> > > On Dec 10, 12:21 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > If so what is the proof that ALL reals belong to that tree?
>
> > There is no proof because the Binary Tree contains all reals between 0
> > and 1 by definition.
>
> What definition is that? I know of no such definition.
All real numbers that have binary representation starting with 0. are
in the Binary Tree and are in the interval [0, 1]. No one is missing
in any of these systems.
> Every binary sequence is in any complete infinite binary tree.
Of course. But you had forgotten that above?
>
> > Cantor's proof (concerning the binaries with bits w and m) shows that
> > not all that are in the tree can be in the list.
>
> Right for once!
> > This proof is given by constructing the whole Binary Tree by countably
> > many actually infinite paths (i.e. finite paths with infinite endings
> > like 010101... or 000... or 111... or 001001001... or any desired
> > ending that I do not publish). My proof rests upon your inability to
> > find out what paths are missing.
>
> And Cantor's counterproof says that no such list contains all paths and
> provides an unambiguous way of determining from any such list ( which
> exists as a consequence of WM's claim of countability) any finite number
> of the uncountably many paths which are missing.
But his proof is taken as evidence that the Binary Tree contains
uncountably many paths *that are defined by nodes only* (as is every
path in the Binary Tree). And this result is false.
> > Find a path that I have not used!
>
> List the ones you have used
I have constructed all paths that can be defined by nodes in the
Binary Tree, because no node is missing in my construction.
> (and until you have listed them , or at
> least prove you can list them, your claim of countability is
> notstablished) and then it is trivial to find others.
Wrong. The notion of countability is self-contradictory. There is no
chance to recognize more than countably many paths by nodes.
Regards, WM
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