Posts:
791
Registered:
9/1/10
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Re: Mathematics in brief
Posted:
Dec 10, 2012 7:24 AM
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On Dec 10, 2:09 am, Zuhair <zaljo...@gmail.com> wrote:
> On Dec 10, 12:17 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > On 10 Dez., 09:07, Virgil <vir...@ligriv.com> wrote:
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> > > 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?
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> > > > 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.
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> > > 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.
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> > > 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 ending that I do not publish).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.
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> > > > Find a path that I have not used!
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> > > List the ones you have used
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> > I have constructed all paths that can be defined by nodes in the
> > Binary Tree, because no node is missing in my construction.
>
> Which Binary Tree, Is it the binary tree on NxN grid? which indeed has
> countably many nodes. If we hold that the number of paths cannot
> exceed the number of nodes, then clearly it follows that this Binary
> tree of yours is countable. But the question is:
>
> Why must we believe that ALL reals can be represented in that
> Countable binary tree of yours.
> Just diagonalize the set of all infinite paths of your binary tree,
> and then you'll get a path that is not in your tree and that of course
> correspond to some real. So your tree provably cannot supply a
> representation for each real.
>
> Zuhair
math
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