Date: Dec 25, 2012 4:22 AM
Author: Virgil
Subject: Re: Simple Refutation of Cantor's Proof

In article 
<9a1af3e6-7c59-43ed-9e47-b4acd3b9eb90@p7g2000pbz.googlegroups.com>,
Graham Cooper <grahamcooper7@gmail.com> wrote:

> On Dec 25, 3:05 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <8526068c-c873-4164-88f9-8717127e3...@ah9g2000pbd.googlegroups.com>,
> >  Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> >
> >
> >
> >
> >
> >
> >
> >

> > > On Dec 25, 4:48 am, William Hughes <wpihug...@gmail.com> wrote:
> > > > On Dec 24, 4:01 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > > > +----->
> > > > > | 0. 542..
> > > > > | 0. 983..
> > > > > | 0. 143..
> > > > > | 0. 543..
> > > > > | ...
> > > > > v

> >
> > > > > OK - THINK
> >
> > > > Induction
> >
> > > only in the HR sense Will!
> >
> > > OK some great replies here!
> >
> > > I think WM point is correct here, the ENTIRE LIST can exist
> >
> > > +----->
> > > | 0. 542..
> > > | 0. 983..
> > > | 0. 143..
> > > | 0. 543..
> > > | ...
> > > v

> >
> > > each digit a FINITE distance from the Origin.
> >
> > > There is no constructible issue with the List itself.
> >
> > > ----------------------
> >
> > > And GG has a good point, once he saw my 2 claims Sequitur together
> >
> > > [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.

The the "anti-diagonal" will differ not only with the nth listed element
at its nth digit, it will also differ from any other representation, to
that base, of the same number, Some numbers in any base will have dual
representations so an "antidiagonal" must be designed to avoid both
whenever there are two.
--