The contardictions of dark numbers found hitherto can be seen here:

https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html

https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html

Regards, WM

On Sunday, September 4, 2022 at 2:41:33 PM UTC+2, WM wrote:

> The contardictions of dark numbers found hitherto can be seen here: [...]

"It's hard to impossible to give a counteragument to a proof that just makes no sense, for example using undefined terms... if I said that "hjhjhjqrt" proves the Riemann hypothesis what would you say?" --David C. Ullrich

"A proof of the existence of an object when the object itself has not yet been defined does not make for a convincing proof." --Matt E.

Yaeh, it's well known: "[WM’s] conclusions are based on the sloppiness of his notions, his inability of giving precise definitions, his fundamental misunderstanding of elementary mathematical concepts, and sometimes, as the late Dik Winter remarked [...], on nothing at all."

On 9/4/2022 7:41 AM, WM wrote:
> The contardictions of dark numbers found hitherto can be seen here:
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>
> Regards, WM

that is a Fake Math book by a known Troll.

On Sunday, 4 September 2022 at 09:41:33 UTC-3, WM wrote:
> The contardictions of dark numbers found hitherto can be seen here:
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html

At least you figured out in the second question that you need to remove the sigfile if you want to present a halfway believable sockpuppet.

söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> The contardictions of dark numbers found hitherto can be seen here:
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>
> Regards, WM

Yes, they are a contradiction, that is why they do not exist.

zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
> söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> > The contardictions of dark numbers found hitherto can be seen here:
> >
> > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> >
> > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> >
> Yes, they are a contradiction, that is why they do not exist.

Nobody could point to a contradiction. Isn't that strange?

Regards, WM

Gus Gassmann schrieb am Sonntag, 4. September 2022 um 16:17:17 UTC+2:
> On Sunday, 4 September 2022 at 09:41:33 UTC-3, WM wrote:
> > The contardictions of dark numbers found hitherto can be seen here:
> >
> > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> >
> > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> At least you figured out in the second question that you need to remove the sigfile if you want to present a halfway believable sockpuppet.

I have no influence on what is published in my name.

Regards, WM

tisdag 6 september 2022 kl. 11:51:19 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
> > söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> > > The contardictions of dark numbers found hitherto can be seen here:
> > >
> > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> > >
> > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> > >
> > Yes, they are a contradiction, that is why they do not exist.
> Nobody could point to a contradiction. Isn't that strange?
>
> Regards, WM
We have pointed it out many times, predominately in that YOU CANNOT DEFINE THEM!

On 9/6/2022 4:51 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
>> söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
>>> The contardictions of dark numbers found hitherto can be seen here:
>>>
>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>>>
>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>>>
>> Yes, they are a contradiction, that is why they do not exist.
>
> Nobody could point to a contradiction. Isn't that strange?

Liar. Everyone in these newsgroups forums has pointed out your self contradictions, and have Proved you wrong many times.

>
> Regards, WM

zelos...@gmail.com schrieb am Dienstag, 6. September 2022 um 14:40:07 UTC+2:
> tisdag 6 september 2022 kl. 11:51:19 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
> > > söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> > > > The contardictions of dark numbers found hitherto can be seen here:
> > > >
> > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> > > >
> > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> > > >
> > > Yes, they are a contradiction, that is why they do not exist.
> > Nobody could point to a contradiction. Isn't that strange?
> >
> We have pointed it out many times, predominately in that YOU CANNOT DEFINE THEM!

First, dark numbers are defined as complements of definable numbers which are defined in
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

But importantly, the matter here are the X's and the O's defined as indexed and not indexed fractions. The X can be shuffled but they will never cover the whole matrix. This fact refutes Cantor but cannot be refuted itself.

Here is it again: k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

Integer fractions are denoted by X's:

XOOO...
XOOO...
XOOO...
XOOO...
....

In the first 3 steps we get the following matrices

XXOO...
OOOO...
XOOO...
XOOO...
....

XXOO...
XOOO...
OOOO...
XOOO...
....

XXXO...
XOOO...
OOOO...
OOOO...
....

and so on.

Regards, WM

On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote: <bla>

Matrix A:

1/1, 1/2, 1/3, ...
2/1, 2/2, 2/3, ...
3/1, 3/2, 3/3, ...
....

Matrix B:

1/1, 2/1, 4/1, ...
3/1, 5/1, 8/1, ...
6/1, 9/1, 13/1, ...
....

What's your problem you psychotic asshole?

On 9/6/2022 4:52 AM, WM wrote:
> Gus Gassmann schrieb am Sonntag, 4. September 2022 um 16:17:17 UTC+2:
>> On Sunday, 4 September 2022 at 09:41:33 UTC-3, WM wrote:
>>> The contardictions of dark numbers found hitherto can be seen here:
>>>
>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>>>
>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>> At least you figured out in the second question that you need to remove the sigfile if you want to present a halfway believable sockpuppet.
>
> I have no influence on what is published in my name.

Wrong. You publish using your name.

>
> Regards, WM

On 9/6/2022 11:36 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 6. September 2022 um 14:40:07 UTC+2:
>> tisdag 6 september 2022 kl. 11:51:19 UTC+2 skrev WM:
>>> zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
>>>> söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
>>>>> The contardictions of dark numbers found hitherto can be seen here:
>>>>>
>>>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>>>>>
>>>>> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>>>>>
>>>> Yes, they are a contradiction, that is why they do not exist.
>>> Nobody could point to a contradiction. Isn't that strange?
>>>
>> We have pointed it out many times, predominately in that YOU CANNOT DEFINE THEM!
>
> First, dark numbers are defined as complements of definable numbers which are defined in
> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

That Troll book is mainly composed of over 157 posts in sci.math, with a closing paragraph by the unfocused yet voluminous Mr Petry

references
John Gabriel,
Graham Cooper,
Anonymous,
D. Christensen,
Bassam King Karzeddin,
David Petry,
Chris M. Thomasson ,
Fred Jeffries,
Zelos Malum, [
Jan Burse
Franz Fritsche,
William Hughes

>

> Regards, WM

tisdag 6 september 2022 kl. 18:36:50 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 6. September 2022 um 14:40:07 UTC+2:
> > tisdag 6 september 2022 kl. 11:51:19 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
> > > > söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> > > > > The contardictions of dark numbers found hitherto can be seen here:
> > > > >
> > > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> > > > >
> > > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> > > > >
> > > > Yes, they are a contradiction, that is why they do not exist.
> > > Nobody could point to a contradiction. Isn't that strange?
> > >
> > We have pointed it out many times, predominately in that YOU CANNOT DEFINE THEM!
> First, dark numbers are defined as complements of definable numbers which are defined in

Which is not defined in any meaningful way!

> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
>
> But importantly, the matter here are the X's and the O's defined as indexed and not indexed fractions. The X can be shuffled but they will never cover the whole matrix. This fact refutes Cantor but cannot be refuted itself.
>
> Here is it again: k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> Integer fractions are denoted by X's:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> In the first 3 steps we get the following matrices
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> ...
>
> XXOO...
> XOOO...
> OOOO...
> XOOO...
> ...
>
> XXXO...
> XOOO...
> OOOO...
> OOOO...
> ...
>
> and so on.
>
> Regards, WM

Which is all IRRELEVANT!

On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 6. September 2022 um 14:40:07 UTC+2:
> > tisdag 6 september 2022 kl. 11:51:19 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Montag, 5. September 2022 um 07:35:10 UTC+2:
> > > > söndag 4 september 2022 kl. 14:41:33 UTC+2 skrev WM:
> > > > > The contardictions of dark numbers found hitherto can be seen here:
> > > > >
> > > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
> > > > >
> > > > > https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
> > > > >
> > > > Yes, they are a contradiction, that is why they do not exist.
> > > Nobody could point to a contradiction. Isn't that strange?
> > >
> > We have pointed it out many times, predominately in that YOU CANNOT DEFINE THEM!
> First, dark numbers are defined as complements of definable numbers which are defined in
> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
>
> But importantly, the matter here are the X's and the O's defined as indexed and not indexed fractions. The X can be shuffled but they will never cover the whole matrix. This fact refutes Cantor but cannot be refuted itself.
>
> Here is it again: k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> Integer fractions are denoted by X's:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> In the first 3 steps we get the following matrices
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> ...
>
> XXOO...
> XOOO...
> OOOO...
> XOOO...
> ...
>
> XXXO...
> XOOO...
> OOOO...
> OOOO...
> ...
>
> and so on.
>
> Regards, WM

--- and so on and on and on and on and on and on and on -----
Mücke, you nonsensical chatter is boring.

On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
>
> The X can [...] cover the whole matrix.

Indeed! Look:

X X X ...
X X X ...
X X X ...
....

You like this ----------->

https://www.youtube.com/watch?v=BecnTRBAZl4

video ?

You know the singer ?

WM kirjutas Pühapäev, 4. september 2022 kl 05:41:33 UTC-7:
> The contardictions of dark numbers found hitherto can be seen here:
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220901%20MO%20Three%20proofs.html
>
> https://www.hs-augsburg.de/~mueckenh/Transfinity/220903%20MSE%20Dark%20numbers%20edited.html
>
> Regards, WM

Fritz Feldhase schrieb am Mittwoch, 7. September 2022 um 09:00:34 UTC+2:
> On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
> > The X can be shuffled but they will never cover the whole matrix.
> > The X can [...] cover the whole matrix.
>
> Indeed! Look:
>
> X X X ...
> X X X ...
> X X X ...
> ...

These X can but these X

XOOO...
XOOO...
XOOO...
XOOO...
....

cannot.

Regards, WM

JVR schrieb am Mittwoch, 7. September 2022 um 08:13:51 UTC+2:
> On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:

> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...

> > XXXO...
> > XOOO...
> > OOOO...
> > OOOO...
> > ...
> >
> > and so on.
> >
> --- and so on and on and on and on and on and on and on -----
> Prof. Dr. Mückenheim, your nonsensical chatter is boring.

So you have not got the newest development?

All mathematicians who have dared to give a sensible answer (contrary to downvoting, cursing, or writing poems) have agreed that the O's can disappear only in the limit or afterwards. This proves that these fractions are not enumerated by Cantor's function
k = (m + n - 1)(m + n - 2)/2 + m
and do not appear in his sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

That is exciting, I'd say, not boring.

Regards, WM

On Thursday, September 8, 2022 at 8:05:45 PM UTC+2, WM wrote:
> JVR schrieb am Mittwoch, 7. September 2022 um 08:13:51 UTC+2:
> > On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
>
> > > XOOO...
> > > XOOO...
> > > XOOO...
> > > XOOO...
> > > ...
> > > XXXO...
> > > XOOO...
> > > OOOO...
> > > OOOO...
> > > ...
> > >
> > > and so on.
> > >
> > --- and so on and on and on and on and on and on and on -----
> > Prof. Dr. Mückenheim, your nonsensical chatter is boring.
>
> So you have not got the newest development?
>
> All mathematicians who have dared to give a sensible answer (contrary to downvoting, cursing, or writing poems) have agreed that the O's can disappear only in the limit or afterwards. This proves that these fractions are not enumerated by Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> and do not appear in his sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
> That is exciting, I'd say, not boring.
>
> Regards, WM

Excitement?

It looks like fear and trembling to me. Why else do you avoid answering
the simple questions:

How did the O's get into your infinite matrix?
If they got in, why can't they get out?
If you can put O's in your matrix why not X's?
Next time, when you manufacture an O-matrix, while you are at it,
just mark position (m,n) with k = (m + n - 1)(m + n - 2)/2 + m.
That would be exciting.
And after that, please drop this idiotic subject.

On Thursday, September 8, 2022 at 7:59:08 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 7. September 2022 um 09:00:34 UTC+2:
> > On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
> > >
> > > The X can [...] cover the whole matrix.
> > >
> > Indeed! Look:
> >
> > X X X ...
> > X X X ...
> > X X X ...
> > ...
> These X can

So we are done. Great. Now look again:

We have infinitely many X_i: X_1, X_2, X_3, ... Actually, the elements in the (infinite) set {X_n : n e IN}.

Now we can distribute them the following way (to fill the first column of a matrix):

X_1 O O ...
X_2 O O ...
X_3 O O ...
...

or we can distribute them the following way (to fill all elements of a matrix):

X_1 X_2 X_4 ...
X_3 X_5 X_8 ...
X_6 X_9 X_1/3 ...
....

Why are you so bad at math, Mückenheim?

WM submitted this idea :
> JVR schrieb am Mittwoch, 7. September 2022 um 08:13:51 UTC+2:
>> On Tuesday, September 6, 2022 at 6:36:50 PM UTC+2, WM wrote:
>
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>
>>> XXXO...
>>> XOOO...
>>> OOOO...
>>> OOOO...
>>> ...
>>>
>>> and so on.
>>>
>> --- and so on and on and on and on and on and on and on -----
>> Prof. Dr. Mückenheim, your nonsensical chatter is boring.
>
> So you have not got the newest development?
>
> All mathematicians who have dared to give a sensible answer (contrary to
> downvoting, cursing, or writing poems) have agreed that the O's can disappear
> only in the limit or afterwards.

They cannot, as you have stipulated that there are infinitely many of
each. Also, your 'swapping' won't change that.

JVR schrieb am Donnerstag, 8. September 2022 um 20:58:03 UTC+2:

> How did the O's get into your infinite matrix?

All matrix elements initially were O's.
Then I indexed the integer fractions and abbreviated the indexed fractions by X's.
> If they got in, why can't they get out?

They can get out, but not in the way Cantor prescribed: Exchanging X and O.
> If you can put O's in your matrix why not X's?

I can put X's but not after the first column is abbreviated by all X's and only Cantor's prescription for distributing the X's is to be followed
> Next time, when you manufacture an O-matrix, while you are at it,
> just mark position (m,n) with k = (m + n - 1)(m + n - 2)/2 + m.
> That would be exciting.

As my proof shows, most fractions cannot be enumerated.

Regards, WM

Fritz Feldhase schrieb am Donnerstag, 8. September 2022 um 21:31:00 UTC+2:

> Now we can distribute them the following way (to fill the first column of a matrix):
>
> X_1 O O ...
> X_2 O O ...
> X_3 O O ...
> ...
>
> or we can distribute them the following way (to fill all elements of a matrix):
>
> X_1 X_2 X_4 ...
> X_3 X_5 X_8 ...
> X_6 X_9 X_1/3 ...
> ...

But the first matrix cannot be turned into the second matrix according to Cantor's prescription, namely by exchanging X and O.

Regards, WM

FromTheRafters schrieb am Donnerstag, 8. September 2022 um 22:43:29 UTC+2:
> WM submitted this idea :

> > All mathematicians who have dared to give a sensible answer (contrary to
> > downvoting, cursing, or writing poems) have agreed that the O's can disappear
> > only in the limit or afterwards.
> They cannot, as you have stipulated that there are infinitely many of
> each. Also, your 'swapping' won't change that.

Of course no O will disappear by swapping. And if in the limit, then the disappearance is not individually definable.

Regards, WM