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