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

> <67261d2b-d57e-4174-af7b-921ac287e...@r4g2000pbi.googlegroups.com>,

> 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

so your claim is

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

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

That's 0.666..

and your claim is

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

is ALSO absent from L(x,y)

That's 0.566..

So every digit you add to the missing string is arbitrary.

Herc