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