Date: Feb 4, 2013 5:14 PM
Author: Virgil
Subject: Re: WMytheology � 203

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 3 Feb., 23:04, fom <fomJ...@nyms.net> wrote:
>

> > > There is no sensible way of saying that 0.111... is more than every
> > > FIS. And every FIS is in a line.

> >
> > Do you mean "is in some line"?

>
> What is s? In decimal it is 1/9. In binary it is 1. In paths or
> strings of bits or decimals it does not exist!

> >
> > As in "there exists a line containing a given FIS"-

>
> There is not more than every FIS of 0.111... Not even all FIS. Here
> you may see why:
> 1) The set of all real numbers of the unit interval is (said to be)
> uncountable.

In ZF, and most other set theories in which the set of all naturals is
allowed, the uncountability of the set of all subsets of the set of
naturals is provable.

> 2) An uncountable set has (infinitely many) more elements than a
> countable set.

Quite so, at least outside WMytheology.

> 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).

But nothing guarantees that one can always find such a decimal or binary
representation.

> 4) In many cases the string cannot be defined by a finite word.

That some reals are inaccessible is well known.

> 5) Without loss of information the first bits of two strings, if
> equal, need not be written twice.

Without loss of information the FISs of uncountably many binary strings,
if equal, need not be written twice.

> 6) Application of this rule leads to the Binary Tree.

Since WM's misunderstandings of of a Complete Infinite Binary Tree are
pervasive and apparently incorrectable, from here on WM's stuff is even
more junky than usual.
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