Date: Dec 25, 2012 7:29 PM
Author: Graham Cooper
Subject: Re: Simple Refutation of Cantor's Proof

On Dec 26, 9:00 am, Virgil <vir...@ligriv.com> wrote:
> In article
>  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
>

> > 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
>
> Since the constructed string must differ from each string listed in L in
> a way that gives it a different value from that listed string, how can
> it possibly still be among a set of values from which it is by
> construction excluded?
>

the size of the string is not fixed.

You construct a portion, you construct another portion, and you never
stop.

Take the Transpose Plane

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

Your construction is now the DIAGONAL of that plane.

0 . T(1,1)  T(2,2)  T(3,3) ...

+----->
| 0. 6 ..
| 0. 5 ..
| 0. 6..
| 0.
| ...
v

0 . T(3,1) T(4,2) T(1,3) ...

is ALSO absent from L(x,y) right?

That's 0.666..