Date: Dec 25, 2012 4:01 PM
Author: Graham Cooper
Subject: Re: Simple Refutation of Cantor's Proof

On Dec 25, 7:22 pm, Virgil <vir...@ligriv.com> wrote:
> > > > [1] you change each digit ONE AT A TIME
> > > > 0.694...
> > > > but this process NEVER STOPS
>
> > > > [2] and you NEVER CONSTRUCT A NEW DIGIT SEQUENCE!
>
> > > Do you deny that f(x) = x^2 and g(x) = 2*x+3 define real functions,
> > > i.e., functions taking arbirary real numbers as arguments and producing
> > > from them appropriate real numbers as values?
>
> > > It you accept them as functions why balk at functions from |N to
> > > the set of decimal digits, interpreted as reals in [0,1]?
>
> > the logical manipulations do not hold on AD(x) = D(x)+1   [mod 9]
>
> > This is what you are really doing.
>
> > +----->
> > | 0. 542..
> > | 0. 983..
> > | 0. 143..
> > | 0. 543..
> > | ...
> > v
>
> > T(x,y) = L(x,y)+1   [mod 9]
>
> > +----->
> > | 0. 653..
> > | 0. 004..
> > | 0. 254..
> > | 0. 654..
> > | ...
> > v
>
> > This plane exists as much as your altered string.
>
> > It's mere naivety to define any digit string from
>
> > 0 . T(1,_)  T(2,_)  T(3,_) ...
>
> > where the set of free values _ biject N
> > and then conclude such strings are absent from L.
>
> > Herc
>
> Since that is not a rule used by anyone who knows what is needed, it is
> irrelevant,
>
> A rule that actually works on decimals, or with any base larger than 7,
> is to look at the nth digit of the nth element in the list and if it
> less than a 6 make the nth digit of the "anti-diagonal a  6 but
> otherwise make it a 5.
>

OK!

+----->
| 0. 542..
| 0. 983..
| 0. 143..
| 0. 543..
| ...
v

T(x,y) = 6  IFF L(x,y) < 6
T(x,y) = 5 OTHERWISE


+----->
| 0. 666..
| 0. 556..
| 0. 666..
| 0. 666..
| ...
v

This plane exists as much as your altered string.
It's mere naivety to define any digit string from

0 . T(1,_)  T(2,_)  T(3,_) ...

where the set of free values _ biject N
and then conclude such strings are absent from L.

Herc