On Friday, September 9, 2022 at 2:01:45 PM UTC+2, WM wrote:
> 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

Ok, Mücke, I see the light.
I was wrong all these many years.
Your deep analysis and imaginative rigorous proofs have convinced me:
Only a complete dunce can believe that
k = (m + n - 1)(m + n - 2)/2 + m
is a biunique mapping between N and NxN.
Just one last question --- oh, never mind, I know the answer already.

On 9/9/2022 7:01 AM, WM wrote:
> 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.

ok, "All matrix elements initially were O's"

O O O O ...
O O O O ...
O O O O ...
O O O O ...
.. . . . ...
.. . . . ...
.. . . . ...

> Then I indexed the *integer fractions* and abbreviated the *indexed fractions* by X's.

ok, now you have;

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

>
>> 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.

no, there is no exchange, you replaced each O with an X, where did you dump all the O bodies ? (at each replacement of an O by an X)

>
>> 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

which results in

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

there is no exchange, you replaced each O with an X, where did you dump all the O bodies ?

>
>> 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.

Your Spoof is a Goof. there is NO MATH here.

>
> Regards, WM

On 9/9/2022 7:08 AM, WM wrote:
> 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

Warning!! intentional use of

*Confusinator Words*

No Ants were harmed in this posting.

On 9/9/2022 7:05 AM, WM wrote:
> 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.

so you are stating that your idea does not work.

>
> Regards, WM

JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:

> Only a complete dunce can believe that
> k = (m + n - 1)(m + n - 2)/2 + m
> is a biunique mapping between N and NxN.

It is a bijection between the potentially infinite collections ℕ_def and ℕ_def x ℕ_def.
For every definable n and m we can find a definable k and vice versa. But every definable k is in finite distance from 0 and has an actual infinity of successors in ℕ.
Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.

Regards, WM

On 9/10/2022 10:13 AM, WM wrote:
> JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:
>
>> Only a complete dunce can believe that
>> k = (m + n - 1)(m + n - 2)/2 + m
>> is a biunique mapping between N and NxN.
>
> It is a bijection between the potentially infinite collections ℕ_def QUACK! and ℕ_def QUACK!! x ℕ_def QUACK!!!.

> For every definable QUACK! n and m we can find a definable QUACK!! k and vice versa. But every definable QUACK!! k is in finite distance from 0 and has an actual infinity of successors in ℕ.

> Only the confusion between potential QUACK!! infinity and actual infinity has resulted in Cantor's set theory.

wrong again. (note QUACK!! in above)

>
> Regards, WM

On Friday, September 9, 2022 at 2:05:33 PM UTC+2, WM wrote:
> 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 [by a sequence of paiwise exchages of] X and O.

Right.

But NO ONE -except you- ever suggested that this might or should be possible.

This is just one of your delusions, you psychotic asshole full of shit.

Hint: ***For everyone else*** it suffices that the second matrix just _exists_.

On Friday, September 9, 2022 at 6:05:17 PM UTC+2, Sergio wrote:

> No Ants were harmed in this posting.

Not sure about this. Maybe one of them read WM's statements...

WM drivels:

> JVR schrieb:

> ... For every definable n

ROTFL ... "definable", if you cannot define things, then
better pack it all in (in German: nimm dir einen Strick)...

On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
> JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:
>
> > Only a complete dunce can believe that
> > k = (m + n - 1)(m + n - 2)/2 + m
> > is a biunique mapping between N and NxN.
> It is a bijection between the potentially infinite collections ℕ_def and ℕ_def x ℕ_def.
> For every definable n and m we can find a definable k and vice versa. But every definable k is in finite distance from 0 and has an actual infinity of successors in ℕ.
> Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.
>
> Regards, WM

I'm very glad to hear that. We are very close to a definition of
'definable' and 'undefinable'.

You are saying that whenever n and m are 'definable',
k = (m + n - 1)(m + n - 2)/2 + m
is also definable.

You understand, I hope, that by induction this makes every integer definable.
So now all we need is a definition of 'undefinable' and we then can proceed to
convince all men of good will that Cantor was a fool.

On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
>
> Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.

A fascinating view! Though not shared by anyone _except you_.

Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:24:35 UTC+2:
> On Friday, September 9, 2022 at 2:05:33 PM UTC+2, WM wrote:
> > 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 [by a sequence of paiwise exchages of] X and O.
>
> Right.
>
> But NO ONE -except you- ever suggested that this might or should be possible.

Cantor has. Enumerating fractions means assigning X's to O's. Enumerating all fractions means assingning X's to all O's. But if we are careful we see that there are not enough X's.

Regards, WM

JVR schrieb am Sonntag, 11. September 2022 um 12:42:47 UTC+2:
> On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
> > JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:
> >
> > > Only a complete dunce can believe that
> > > k = (m + n - 1)(m + n - 2)/2 + m
> > > is a biunique mapping between N and NxN.
> > It is a bijection between the potentially infinite collections ℕ_def and ℕ_def x ℕ_def.
> > For every definable n and m we can find a definable k and vice versa. But every definable k is in finite distance from 0 and has an actual infinity of successors in ℕ.
> > Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.
> >
> I'm very glad to hear that. We are very close to a definition of
> 'definable' and 'undefinable'.
>
> You are saying that whenever n and m are 'definable',
> k = (m + n - 1)(m + n - 2)/2 + m
> is also definable.
>
> You understand, I hope, that by induction this makes every integer definable.

That results in a potentially infinite collection. By induction we prove that in actual infinity almost all natnumbers are undefinable.

If n has ℵ₀ successors, then n+1 has ℵ₀ successors.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

> So now all we need is a definition of 'undefinable'

Simplest among many equivalent definitions: A natural number is undefined if no FISON is known. A natural number is undefinable if no FISON can be known.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 11. September 2022 um 13:56:04 UTC+2:
> On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
> >
> > Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.
> A fascinating view! Though not shared by anyone _except you_.

An infinite set according to Cantor is complete. That means nothing can be added. Hilbert's hotel adds another room with no problems. That shows it is not complete. Potential infinity.

Regards, WM

On 9/11/2022 5:42 AM, JVR wrote:
> On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
>> JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:
>>
>>> Only a complete dunce can believe that
>>> k = (m + n - 1)(m + n - 2)/2 + m
>>> is a biunique mapping between N and NxN.
>> It is a bijection between the potentially infinite collections ℕ_def and ℕ_def x ℕ_def.
>> For every definable n and m we can find a definable k and vice versa. But every definable k is in finite distance from 0 and has an actual infinity of successors in ℕ.
>> Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.
>>
>> Regards, WM
>
> I'm very glad to hear that. We are very close to a definition of
> 'definable' and 'undefinable'.
>
> You are saying that whenever n and m are 'definable',
> k = (m + n - 1)(m + n - 2)/2 + m
> is also definable.
>
> You understand, I hope, that by induction this makes every integer definable.
> So now all we need is a definition of 'undefinable' and we then can proceed to
> convince all men of good will that Cantor was a fool.

NEW " definable" ANTS!!

Please provide further dafination of the following definables...

non definable Ants
re definable Ants
Co definable Ants
indefinable Ants
antidefinable Ants
Contradefinable Ants
exdefinable Ants
autodefinable Ants
exodefinable Ants
extradefinable Ants
hyperdefinable Ants
Interdefinable Ants
microdefinable Ants
overdefinable Ants
postdefinable Ants
predefinable Ants
semidefinable Ants
Subdefinable Ants
Transdefinable Ants
misdefinable Ants
megadefinable Ants

On Sunday, 11 September 2022 at 10:14:28 UTC-3, WM wrote:
[...]
> An infinite set according to Cantor is complete. That means nothing can be added. Hilbert's hotel adds another room [...]

No room is added, you imbecile! You truly have *NO CLUE* about infinity.

On Sunday, September 11, 2022 at 3:05:21 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:24:35 UTC+2:
> > On Friday, September 9, 2022 at 2:05:33 PM UTC+2, WM wrote:
> > > 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 [by a sequence of pairwise exchages of] Xs and Os.
> > >
> > Right.
> >
> > But NO ONE -except you- ever suggested that this might or should be possible.
> >
> Cantor has.

No, he hasn't.

Mind to present a quote to back up your (false) claim?

On Sunday, September 11, 2022 at 3:14:28 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 11. September 2022 um 13:56:04 UTC+2:
> > On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
> > >
> > > Only the confusion between potential infinity and actual infinity has resulted in Cantor's set theory.
> > >
> > A fascinating view! Though not shared by anyone _except you_.
> >
> An infinite set according to Cantor is complete. That means nothing can be added.

Indeed! You CAN'T "add" anything _to any set_, since sets dont grow [or shrink].

On the other hand, we may consider the sets IN and IN' := IN u {omega}. Hence IN' contains ALL elements in IN _and_ (just) omega (as elements). This means IN' \ IN = {omega}.

> Hilbert's hotel adds another room with no problems.

Nonsense. You are just making this up, you silly crank!!!

No room is added. Seems you don't know (any more?) what you are talking about!

WM laid this down on his screen :
> JVR schrieb am Sonntag, 11. September 2022 um 12:42:47 UTC+2:
>> On Saturday, September 10, 2022 at 5:13:48 PM UTC+2, WM wrote:
>>> JVR schrieb am Freitag, 9. September 2022 um 16:03:10 UTC+2:
>>>
>>>> Only a complete dunce can believe that
>>>> k = (m + n - 1)(m + n - 2)/2 + m
>>>> is a biunique mapping between N and NxN.
>>> It is a bijection between the potentially infinite collections ℕ_def and
>>> ℕ_def x ℕ_def. For every definable n and m we can find a definable k and
>>> vice versa. But every definable k is in finite distance from 0 and has an
>>> actual infinity of successors in ℕ. Only the confusion between potential
>>> infinity and actual infinity has resulted in Cantor's set theory.
>>>
>> I'm very glad to hear that. We are very close to a definition of
>> 'definable' and 'undefinable'.
>>
>> You are saying that whenever n and m are 'definable',
>> k = (m + n - 1)(m + n - 2)/2 + m
>> is also definable.
>>
>> You understand, I hope, that by induction this makes every integer
>> definable.
>
> That results in a potentially infinite collection. By induction we prove that
> in actual infinity almost all natnumbers are undefinable.
>
> If n has ℵ₀ successors, then n+1 has ℵ₀ successors.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>
>> So now all we need is a definition of 'undefinable'
>
> Simplest among many equivalent definitions: A natural number is undefined if
> no FISON is known. A natural number is undefinable if no FISON can be known.

That doesn't help.

On Sunday, September 11, 2022 at 8:50:22 PM UTC+2, FromTheRafters wrote:
> WM laid this down on his screen :
> >
> > Simplest among many equivalent definitions: A natural number is undefined if
> > no FISON is known. A natural number is undefinable if no FISON can be known.
> >
> That doesn't help.

No? Why not? Imho these are great definitions!!!1!!

"A natural number is undefined if no FISON is known."

"A natural number"? Any? A certain? Which one?

Ok, so let's just plug in a variable, say n:

"A natural number n is undefined if no FISON is known."

Again, I'm not quite certain... Any natural number n? A certain number n? Which one?

Let's try try the following one - to be more specific:

"The natural number n is undefined if no FISON is known."

And let't try it with n = 1, say:

"The natural number 1 is undefined if no FISON is known."

Hmmm... But we KNOW a FISON, for example the FISON {1, 2, 3}. :-)

Hence 1 is not not undefined (I'd say), and hence _defined_.

Same with n = 2, 3, 4, ... after all, we KNOW a FISON! :-)

Hence each and every natural number is _defined_.

Moreover this means that a FISON _can be knows_ after all, we _know_ one!

Hence each and every natural number is _definable_ too!

On the other hand, no big surprise...

On 9/11/2022 8:05 AM, WM wrote:
> Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:24:35 UTC+2:
>> On Friday, September 9, 2022 at 2:05:33 PM UTC+2, WM wrote:
>>> 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 [by a sequence of paiwise exchages of] X and O.
>>
>> Right.
>>
>> But NO ONE -except you- ever suggested that this might or should be possible.
>
> Cantor has.

Liar.

>Enumerating fractions means assigning X's to O's.

Liar.

>Enumerating all fractions means assingning X's to all O's.

Liar.

> But if we are careful we see that there are not enough X's.

you need to get a new cheese for your cracker.

>
> Regards, WM
>

Fritz Feldhase laid this down on his screen :
> On Sunday, September 11, 2022 at 8:50:22 PM UTC+2, FromTheRafters wrote:
>> WM laid this down on his screen :
>>>
>>> Simplest among many equivalent definitions: A natural number is undefined
>>> if no FISON is known. A natural number is undefinable if no FISON can be
>>> known.
>>>
>> That doesn't help.
>
> No? Why not? Imho these are great definitions!!!1!!
>
> "A natural number is undefined if no FISON is known."
>
> "A natural number"? Any? A certain? Which one?
>
> Ok, so let's just plug in a variable, say n:
>
> "A natural number n is undefined if no FISON is known."
>
> Again, I'm not quite certain... Any natural number n? A certain number n?
> Which one?
>
> Let's try try the following one - to be more specific:
>
> "The natural number n is undefined if no FISON is known."
>
> And let't try it with n = 1, say:
>
> "The natural number 1 is undefined if no FISON is known."
>
> Hmmm... But we KNOW a FISON, for example the FISON {1, 2, 3}. :-)

I got more if you need them.

> Hence 1 is not not undefined (I'd say), and hence _defined_.
>
> Same with n = 2, 3, 4, ... after all, we KNOW a FISON! :-)
>
> Hence each and every natural number is _defined_.

Whew!

> Moreover this means that a FISON _can be knows_ after all, we _know_ one!

Yeah, his imprecise language always seems to defeat him.

> Hence each and every natural number is _definable_ too!
>
> On the other hand, no big surprise...

Indeed.

Gus Gassmann schrieb am Sonntag, 11. September 2022 um 15:41:29 UTC+2:
> On Sunday, 11 September 2022 at 10:14:28 UTC-3, WM wrote:
> [...]
> > An infinite set according to Cantor is complete. That means nothing can be added. Hilbert's hotel adds another room [...]
>
> No room is added

A person is added. A room is occupied that before was not occupied.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 11. September 2022 um 16:39:31 UTC+2:
> On Sunday, September 11, 2022 at 3:05:21 PM UTC+2, WM wrote:
> > Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:24:35 UTC+2:
> > > On Friday, September 9, 2022 at 2:05:33 PM UTC+2, WM wrote:
> > > > 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 [by a sequence of pairwise exchages of] Xs and Os.
> > > >
> > > Right.
> > >
> > > But NO ONE -except you- ever suggested that this might or should be possible.
> > >
> > Cantor has.
> No, he hasn't.
>
> Mind to present a quote to back up your (false) claim?

"wenn sie sich gegenseitig eindeutig, Element für Element einander zuordnen lassen." [Cantor, p. 167]

Regards, WM

Fritz Feldhase schrieb am Sonntag, 11. September 2022 um 16:48:48 UTC+2:
> On Sunday, September 11, 2022 at 3:14:28 PM UTC+2, WM wrote:

> > Hilbert's hotel adds another room with no problems.
> Nonsense. You are just making this up, you silly crank!!!
>
> No room is added.

Every person has his own room. A person is added. Every person has his own room.

Regards, WM