Date: Feb 23, 2013 11:21 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 22 Feb., 23:23, Virgil <vir...@ligriv.com> wrote:

> > Here is a summary of the argument concerning the Binary Tree:
>
> > 1) The set of all real numbers of the unit interval is (said to be)
> > uncountable.
> > 2) An uncountable set has (infinitely many) more elements than a
> > countable set.
> > 3) Every real number has at least one unique representation as an
> > infinite binary string (some rationals have even two representations
> > but that's peanuts).
> > 4) In many cases the string cannot be defined by a finite word.
> > 5) Without loss of information the first bits of two strings, if
> > equal, need not be written twice.
> > 6) Application of this rule leads to the Binary Tree.
> > 7) The binary strings of the unit interval are isomorphic to the paths
> > of the Binary Tree.

>
> If WM means they are of equal cardinality or biject with each other ,
> true, but to establish an isomorphism, as WM is claiming, one must
> specify the structure that is being preserved by the bijection, which WM
> has NOT done.


The mapping is bijective and linear.

> > 8) It is not possible to distinguish more than countably many paths by
> > their nodes.

>
> The set of paths of a CIBT is easily bijected with the set of all
> subsets of |N (the path generates the set of naturals corresponding to
> the levels at which that path branches left rather than right) which
> allows us easily to distinguish any path from any other by the diffences
> in their corresponding sets of naturals.


This shows a contradiction - at least in case someone accepts
Hessenberg's trick as part of mathematics.

Regards, WM