A Non-Row Diagonal

[Graham Cooper]

THEOREM 1: A sort algorithm exists for any non-trivially-incomplete
list of reals such that the diagonal of the sorted list is not equal
to any row of the list.

This diagonal will be called a Non-Row Diagonal.

e.g.

L
0.12345..
0.22222..
0.33333..
0.44444..
..

Start with L[1,1] = 1

Try L[2,2] = 2

D = 0.12...

This is equal to the prefix of ROW 1. 0.12345...

So swap row 2 & 3

0.[1]2345..
0.3[3]333..
0.22222..
0.44444..
..

D = 0.13...
which does not equal ROW1 or ROW2

For any nXn top left section of the List
There are 10^n possible prefix digit strings
and only n digit strings

So it is possible to guarantee for any length n of the diagonal D.

prefix(D, n) =/= prefix(r1, n)
prefix(D, n) =/= prefix(r2, n)
...
prefix(D, n) =/= prefix(rn, n)

while constructing D for n=1,2,3...

As n->oo this ensures the diagonal D does not equal any row.

---

THEOREM 2: A sort algorithm exists for any non-trivially-incomplete
list of reals such that the diagonal of the sorted list is not equal
to any given Non-Row Diagonal of the list for any digit.

This diagonal is also a Non-Row Diagonal

i.e.

A(non-row-diagonal-1):
E(non-row-diagonal-2):
ANTI(non-row-diagonal-1) = non-row-diagonal-2

That is, assuming the diagonal is not some row of the list,
then changing the diagonal only has the effect of sorting the
list to have the changed diagonal (as it's diagonal!)


0. [0] 100010 ..
0. 100 [1] 000..
0. 1 [1] 11000..
0. 00 [0] 1100.. ----> 0.0101...
..
0. 010 [0] 010.. ----> 0.1010...
0. [1] 010000 ..
0. 11 [1] 1000..
0. 0 [0] 01100..
..

Changing every digit of the diagonal...
into a NEW DIAGONAL




Herc

[christian.bau]

On Nov 26, 6:24 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> THEOREM 1: A sort algorithm exists for any non-trivially-incomplete
> list of reals such that the diagonal of the sorted list is not equal
> to any row of the list.

And what is the significance of that?

And what would be a "non-trivially-incomplete list of reals", when it
has been known since Cantor that every list of reals is incomplete?

[Graham Cooper]

On Nov 26, 11:54 pm, "christian.bau"

+------------>
! 0 . [0]1 2...
! 0 . 0[1]1...
! 0 . 1 0[2]... -------012
! 0 . 1 2 0...
! 0 . 2 0 1...
! 0 . 2 1 0...
! 0 . 2 2[1]... -------021
! 0 . 1[2]1...
! 0 . [0]0 0...
! 0 . 1 1 0...
! 0 . 1 2 1...
! 0 . 0 2 0...
! 0 . [1] 1 2...
! 0 . 0[0]0...
! 0 . 2 1[2]... -------102
! 0 . 2 2 2...
! 0 . 1 1 1...
! 0 . 0 0 0...
! 0 . [1] 1 0...
! 0 . 1[2]1...
! 0 . 0 2[0]... -------120
! 0 . 1 1 2...
! 0 . 1 20...
! 0 . 2[0]1...
! 0 . [2]1 0...
! 0 . 2 2 2...
! 0 . 1 1[1]... -------201
! 0 . 0 0 0...
! 0 . 1 1 0...
! 0 . 1 2 1...
! 0 . 0 2 0...
! 0 . 1 1 2...
! 0 . 0 2 0...
! 0 . [2]1 0...
! 0 . 2 2 2...
! 0 . 1[1]1...
! 0 . 1 2[0]... -------210
! 0 . 2 0 1...
! 0 . 2 1 0...
! 0 . 2 2 2...
! 0 . 1 1 1...
! 0 . 0[0]0...
! 0 . 1 1[0]... -------000
! 0 . 1 2 1...
! 0 . [0]2 0...
! 0 . 1 1 2...
! 0 . [0]2 0...
! 0 . 2[1]0...
! 0 . 2 2 2...
! 0 . 1 1[1]... -------011
! 0 . 1 2 0...
! 0 . 2 0[1]... -------111
! 0 . 2[1]0...
! 0 . 2 2 2...
! 0 . [1]1 1...
! 0 . 0 0 0...
! 0 . 1 1 0...
! 0 . 1 2 1...
! 0 . 0 2 0...
! 0 . 1 1 2...
! 0 . 0 2[0]... -------110
! 0 . 2[1]0...
! 0 . 2 2 2...
! 0 . [1]1 1...
! 0 . 1 2 0...
! 0 . 2 0 1...
! 0 . 2 1 0...
! 0 . 2 2[2]... -------022
! 0 . 1 1 1...
! 0 . [0]0 0...
! 0 . 1 1 0...
! 0 . 1[2]1...
! 0 . 0 2 0...
! 0 . 1 1[2]... -------222
! 0 . 0[2]0...
! 0 . [2]1 0...
! 0 . 2 2 2...
! 0 . 1 1 1...
! 0 . 0 0 0...
v
...


Herc

[christian.bau]

And how would that answer my two questions?

[Graham Cooper]

On Nov 27, 7:52 am, "christian.bau" <christian....@cbau.wanadoo.co.uk>

It's not trivially incomplete because you cannot calculate an anti-
diagonal without realising your construction process is flawed.

1 & 2!

Herc

[christian.bau]

On Nov 26, 10:07 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> It's not trivially incomplete because you cannot calculate an anti-
> diagonal without realising your construction process is flawed.

You'll need to explain that. We are talking about an infinite matrix,
and there is no such thing as the antidiagonal of an infinite matrix.

[Barb Knox]

In article
<1cfc7aec-72ee-4f29...@u5g2000vbd.googlegroups.com>,

You apparently allow infinite matrices in your mathematics. Therefore
you must surely also allow infinite sequences, since every row of an
infinite matrix is an infinite sequence. Since you allow infinite
sequences which are rows, you must surely also allow infinite sequences
which are diagonals. If you allow infinite diagonal sequences, how can
you possibly disallow infinite anti-diagonal sequences?

--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------

[Graham Cooper]

On Nov 28, 11:47 am, Barb Knox <s...@sig.below> wrote:
> In article
> <1cfc7aec-72ee-4f29-ae67-78c9e3261...@u5g2000vbd.googlegroups.com>,

If you LIST (enumerate) a countable SET

then you establish a diagonal.

For the purposes of what a countable set contains,

the enumeration is insignificant.

The set of women I have conquered is equally

LIST
Jay
Greta
Larah
Dana
Susan
Celeste
Heather
Holly

as it is

LIST
Holly
Heather
Celeste
Susan
Dana
Larah
Greta
Jay



A Diagonal is no different to an Infinite Walk.

And an Infinite Walk (touching every row once in any order,
from digit position 1, 2, 3..)

cannot be relied upon to calculate a new digit string.

Flipping the digits just makes a minor adjustment in the infinite walk
to a near neighbouring row!

If Cantor's Proof was valid, don't you think someone in sci.math could
actually post the anti-diagonal to some of my counterexamples?



A Purported Complete List of All Reals

[Virgil]

In article <see-04D513.1...@news.eternal-september.org>,

Barb Knox <s...@sig.below> wrote:

> In article
> <1cfc7aec-72ee-4f29...@u5g2000vbd.googlegroups.com>,
> "christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:
>
> > On Nov 26, 10:07 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > It's not trivially incomplete because you cannot calculate an anti-
> > > diagonal without realising your construction process is flawed.
> >
> > You'll need to explain that. We are talking about an infinite matrix,
> > and there is no such thing as the antidiagonal of an infinite matrix.
>
> You apparently allow infinite matrices in your mathematics. Therefore
> you must surely also allow infinite sequences, since every row of an
> infinite matrix is an infinite sequence. Since you allow infinite
> sequences which are rows, you must surely also allow infinite sequences
> which are diagonals. If you allow infinite diagonal sequences, how can
> you possibly disallow infinite anti-diagonal sequences?

They do it by disallowing logic, just like WM does.
--

[Graham Cooper]

On Nov 28, 1:19 pm, Virgil <vir...@ligriv.com> wrote:
> In article <see-04D513.14472028112...@news.eternal-september.org>,

>  Barb Knox <s...@sig.below> wrote:
>
>
>
>
>
>
>
>
>
> > In article

> > <1cfc7aec-72ee-4f29-ae67-78c9e3261...@u5g2000vbd.googlegroups.com>,

> >  "christian.bau" <christian....@cbau.wanadoo.co.uk> wrote:
>
> > > On Nov 26, 10:07 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > > It's not trivially incomplete because you cannot calculate an anti-
> > > > diagonal without realising your construction process is flawed.
>
> > > You'll need to explain that. We are talking about an infinite matrix,
> > > and there is no such thing as the antidiagonal of an infinite matrix.
>
> > You apparently allow infinite matrices in your mathematics.  Therefore
> > you must surely also allow infinite sequences, since every row of an
> > infinite matrix is an infinite sequence.  Since you allow infinite
> > sequences which are rows, you must surely also allow infinite sequences
> > which are diagonals.  If you allow infinite diagonal sequences, how can
> > you possibly disallow infinite anti-diagonal sequences?
>
> They do it by disallowing logic, just like WM does.
> --

Right!

You think SIZE(R) > oo

because you allow the logic FORALL(F) F:N->R

in 1st Order Logic!

We don't!

Herc

[Bear Bottoms]

myra...@stiffsrus.com (Sycho) wrote in news:4ed1013b.5718562
@cometothedarkside.com:

> This just in to the alt.2600 news room. On Sat, 26 Nov 2011 05:54:01
> -0800 (PST) it was announced to all in a public briefing,
> "christian.bau" <christ...@cbau.wanadoo.co.uk> made the following
> declaration and shocked the world when the following was announced:

> Wow.. just.. wow. That's deep. I thought the exact same thing too.
>
> Ok, I was lying. But I wore my best poke her face. ;D

Math and science are probably almost 100% based on salesmanship. There
may be 5%, 6% by mistake that's not salesmanship.

--
Bear
http://bearware.info

[christian.bau]

On Nov 28, 1:47 am, Barb Knox <s...@sig.below> wrote:

> You apparently allow infinite matrices in your mathematics.  Therefore
> you must surely also allow infinite sequences, since every row of an
> infinite matrix is an infinite sequence.  Since you allow infinite
> sequences which are rows, you must surely also allow infinite sequences
> which are diagonals.  If you allow infinite diagonal sequences, how can
> you possibly disallow infinite anti-diagonal sequences?

I'm not disallowing anything; the definition of "anti-diagonal" just
doesn't make any sense for an infinite matrix. What would be the first
element of that anti-diagonal?

[Barb Knox]

In article
<6e2cd7a7-136f-4283...@by4g2000vbb.googlegroups.com>,

For base ten fractions, if d is the first digit of the diagonal (i.e.,
the upper left digit in the matrix) then d+5 mod 10 would be the first
digit of the anti-diagonal. (Functions other than d+5 mod 10 for
mapping diagonal digit values to their corresponding anti-diagonal ones
would of course be OK too.)

(This is so basic that I expect I'm not addressing your actual
confusion.)

[Virgil]

In article
<6e2cd7a7-136f-4283...@by4g2000vbb.googlegroups.com>,
"christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:

Since the diagonal elements of that matrix, even when infinite, consists
of those element having row-index and column index equal to each other,
all one needs for its anti-diagonal is a sequence whose nth member
differs from the (n,n)-entry in the matrix,
--

[Virgil]

In article <see-802587.1...@news.eternal-september.org>,

Barb Knox <s...@sig.below> wrote:

> In article
> <6e2cd7a7-136f-4283...@by4g2000vbb.googlegroups.com>,
> "christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:
>
> > On Nov 28, 1:47 am, Barb Knox <s...@sig.below> wrote:
> >
> > > You apparently allow infinite matrices in your mathematics.  Therefore
> > > you must surely also allow infinite sequences, since every row of an
> > > infinite matrix is an infinite sequence.  Since you allow infinite
> > > sequences which are rows, you must surely also allow infinite sequences
> > > which are diagonals.  If you allow infinite diagonal sequences, how can
> > > you possibly disallow infinite anti-diagonal sequences?
> >
> > I'm not disallowing anything; the definition of "anti-diagonal" just
> > doesn't make any sense for an infinite matrix. What would be the first
> > element of that anti-diagonal?
>
> For base ten fractions, if d is the first digit of the diagonal (i.e.,
> the upper left digit in the matrix) then d+5 mod 10 would be the first
> digit of the anti-diagonal. (Functions other than d+5 mod 10 for
> mapping diagonal digit values to their corresponding anti-diagonal ones
> would of course be OK too.)
>
> (This is so basic that I expect I'm not addressing your actual
> confusion.)

Even better would be a rule that would avoid ever using either a '0' or
a '9', as that would prevent the anti-diagnal from matching any real
number having dual decimal representations, like .1000... and
.0999...
--

[Barb Knox]

In article
<5d2cee38-df15-4999...@g1g2000pri.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

> On Nov 28, 11:47 am, Barb Knox <s...@sig.below> wrote:
> > In article
> > <1cfc7aec-72ee-4f29-ae67-78c9e3261...@u5g2000vbd.googlegroups.com>,
> >
> >  "christian.bau" <christian....@cbau.wanadoo.co.uk> wrote:
> > > On Nov 26, 10:07 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > > It's not trivially incomplete because you cannot calculate an anti-
> > > > diagonal without realising your construction process is flawed.
> >
> > > You'll need to explain that. We are talking about an infinite matrix,
> > > and there is no such thing as the antidiagonal of an infinite matrix.
> >
> > You apparently allow infinite matrices in your mathematics.  Therefore
> > you must surely also allow infinite sequences, since every row of an
> > infinite matrix is an infinite sequence.  Since you allow infinite
> > sequences which are rows, you must surely also allow infinite sequences
> > which are diagonals.  If you allow infinite diagonal sequences, how can
> > you possibly disallow infinite anti-diagonal sequences?

> If you LIST (enumerate) a countable SET
> then you establish a diagonal.
>
> For the purposes of what a countable set contains,
> the enumeration is insignificant.

I am relaxing my usual practice of not (knowingly) replying to you,
because here you actually make a sensible mathematical point.

We agree that sets are unordered, so a given countable set S does not
come with any intrinsic enumeration.

BUT (and here comes some maths), consider what it *means* for a set S to
be countable. It means that there is at least one bijection between the
set of natural numbers and S. So, let f be any one of these bijections
(it does not matter which one). The bijection f imposes an enumeration
on S, namely f(0),f(1),f(2),f(3),...

So, given any countable set, there is at least one enumeration of it (in
fact there are vastly many). Let S be any countable set of digit
strings representing real fractions between 0 and 1. Use any one of the
enumerations of S to construct an anti-diagonal digit string. That
string will not be in S.

Different enumerations will produce different anti-diagonals, but the
key fact is that *every* enumeration produces an anti-diagonal. So any
enumeration suffices to generate an anti-diagonal digit string which is
not in S.

[snip]

[Graham Cooper]

On Nov 29, 12:30 pm, Barb Knox <s...@sig.below> wrote:
> In article

> <5d2cee38-df15-4999-a052-c80a368a4...@g1g2000pri.googlegroups.com>,

Yes I get that.

But a random infinite walk is *slightly* different to a randomly
selected permutation.

Here is your argument.

Barb:
oo-strings are well defined for rows
oo-strings are well defined for diagonals
oo-strings are well defined for anti-diagonals

Herc: What about for infinite walks?

Barb: It works ForAll(Lists)... in FOL... who cares... Transfinity is
real!

SIZE(R) > oo



Herc

[Phil Carmody]

Barb Knox <s...@sig.below> writes:
> In article
> <6e2cd7a7-136f-4283...@by4g2000vbb.googlegroups.com>,
> "christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:
>
> > On Nov 28, 1:47 am, Barb Knox <s...@sig.below> wrote:
> >
> > > You apparently allow infinite matrices in your mathematics.  Therefore
> > > you must surely also allow infinite sequences, since every row of an
> > > infinite matrix is an infinite sequence.  Since you allow infinite
> > > sequences which are rows, you must surely also allow infinite sequences
> > > which are diagonals.  If you allow infinite diagonal sequences, how can
> > > you possibly disallow infinite anti-diagonal sequences?
> >
> > I'm not disallowing anything; the definition of "anti-diagonal" just
> > doesn't make any sense for an infinite matrix. What would be the first
> > element of that anti-diagonal?
>
> For base ten fractions, if d is the first digit of the diagonal (i.e.,
> the upper left digit in the matrix) then d+5 mod 10 would be the first
> digit of the anti-diagonal. (Functions other than d+5 mod 10 for
> mapping diagonal digit values to their corresponding anti-diagonal ones
> would of course be OK too.)
>
> (This is so basic that I expect I'm not addressing your actual
> confusion.)

Half right, you were being asked for the last element in the first row
of the original "infinite matrix".

However, also half wrong, as it's not Christian who is confused.
The anti-diagonal would be the diagonal in the opposite orientation,
from top right to bottom left.

Phil
--
Unix is simple. It just takes a genius to understand its simplicity
-- Dennis Ritchie (1941-2011), Unix Co-Creator