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Topic: UNCOUNTABILITY
Replies: 59   Last Post: Dec 24, 2012 2:06 PM

 Messages: [ Previous | Next ]
 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Re: UNCOUNTABILITY
Posted: Dec 22, 2012 7:26 AM

On Dec 22, 11:09 am, Virgil <vir...@ligriv.com> wrote:
> In article
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>  Zuhair <zaljo...@gmail.com> wrote:

> > On Dec 21, 8:44 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 21 Dez., 17:36, Zuhair <zaljo...@gmail.com> wrote:
>
> > > Note finally: Every Cantor diagonal r differs from any other real
> > > number by a finite initial segment n(r) of its string of digits. That
> > > is not possible with the Binary Tree. A diagonal does not differ from
> > > all finite paths, i.e., for every initial segment n(r) of every real
> > > number r there exists a finite path of the Binary Tree that is n(r).
> > > You may consider actual infinity as well as uncountable languages, but
> > > that does not change the fact that Cantor's argument does not apply.

>
> > NO, Cantor's diagonal argument construct a diagonal r that differs
> > from every element of a COUNTABLE set of reals by a finite initial
> > segment n(r). You are just not getting Cantor's argument. Cantor's
> > argument is not about diagonalizing the set of ALL reals since that is
> > clearly not possible, Cantor's argument is about diagonalizing any
> > COUNTABLE set of reals. So again for any *countable* set S of reals
> > there is a diagonal r that differs from each real in S by a finite
> > initial segment n(r).
> > This has been PROVED by Cantor. This logically entails that the set R
> > of ALL reals cannot be countable, since otherwise this leads to the
> > obvious contradiction that R is missing a member of it which cannot
> > be.

>
> > The complete infinite binary tree have paths that can represent any
> > real! So no diagonal path can be defined over the whole set, this is
> > clear. But again also using Cantor's argument we can prove that for
> > Any subtree T of the complete infinite binary tree if T has countably
> > many paths then we can define a diagonal path that is missing from T,
> > i.e. a diagonal path that belongs to the complete infinite binary tree
> > but yet missing from T. Thus the infinite binary tree itself cannot be
> > countable!

>
> > I reviewed your writings about the infinite binary tree. You want to
> > prove that if we assume completed infinity (which you don't believe it
> > to be a consistent assumption, so you don't believe even in the
> > possible existence of such objects that are completed infinite sets)
> > then we will arrive at a contradiction, this contradiction is the
> > Cantor-WM contradiction, that is:

>
> > If we assume that the sets N and R of all naturals and reals
> > respectively are completed infinite sets, then it follows that

>
> > (1) R is strictly bigger than N by Cantor's proof
> > (2) R is as big as N by the WM infinite binary tree proof

>
> I think you meant to say
>
> (2) R is no bigger than N by the WM infinite binary tree proof'
> What you actually said would, at least in standard English, allow R and
> N still to satisfy (1).
>

Note 'strictly' in (1).

Zuhair

Date Subject Author
12/19/12 george
12/20/12 Zaljohar@gmail.com
12/20/12 Barb Knox
12/20/12 ross.finlayson@gmail.com
12/20/12 Zaljohar@gmail.com
12/20/12 ross.finlayson@gmail.com
12/20/12 Graham Cooper
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/20/12 Graham Cooper
12/20/12 mueckenh@rz.fh-augsburg.de
12/20/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/21/12 William Hughes
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 William Hughes
12/22/12 Virgil
12/22/12 William Hughes
12/22/12 Graham Cooper
12/23/12 Virgil
12/23/12 William Hughes
12/24/12 Graham Cooper
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 William Hughes
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/23/12 Virgil
12/22/12 Virgil
12/21/12 Zaljohar@gmail.com
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 William Hughes
12/22/12 Virgil
12/21/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/22/12 Zaljohar@gmail.com
12/22/12 Virgil
12/22/12 Zaljohar@gmail.com
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 Virgil
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 Virgil
12/21/12 Virgil
12/20/12 Virgil
12/20/12 Graham Cooper
12/20/12 Graham Cooper
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/21/12 Graham Cooper
12/21/12 Graham Cooper