Date: Dec 25, 2012 6:00 PM
Author: Virgil
Subject: Re: Simple Refutation of Cantor's Proof

In article 
<67261d2b-d57e-4174-af7b-921ac287e30d@r4g2000pbi.googlegroups.com>,
Graham Cooper <grahamcooper7@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?
--