Gus Gassmann schrieb am Mittwoch, 23. November 2022 um 13:05:52 UTC+1:
> On Tuesday, 22 November 2022 at 18:11:39 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 22. November 2022 um 22:55:22 UTC+1:
> > > On Tuesday, 22 November 2022 at 15:51:31 UTC-4, WM wrote:
> > >
> > > ...nothing
> > that you could discuss without showing that even your inappropriate limits will not help. A clearer contradiction of your claim is hardly possible:
> >
> > If the infinitely many O's leave in the limit, then there is no indexing possible. If they do not leave, then there is no completed indexing.
> That bullshit only shows that you have no idea how limits work.

Exchanging the X's and O's will delete all O's in a definable way but only after all definable terms of the sequence have kept all and every O. You believe that limits work in this way?

Regards, WM

FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
> WM brought next idea :
> > FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
> >
> >> He has imagined a 'set' of numbers whose elements are included in the
> >> set by the criterion that they have no criterion -- and thus are dark.
> >>
> >> Genius, pure and simple.
> >>
> >> Too bad they are useless.
> >
> > They are not useless but useful to prove that the idea of equivalent infinite
> > stes is wrong.
> Of course it is wrong, the naturals with zero and the naturals without
> zero are not the same set, they are the same size though.

Two sets are said to be equivalent if they have the same cardinality.

Regards, WM

On 11/23/2022 9:46 AM, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 23. November 2022 um 13:05:52 UTC+1:
>> On Tuesday, 22 November 2022 at 18:11:39 UTC-4, WM wrote:
>>> Gus Gassmann schrieb am Dienstag, 22. November 2022 um 22:55:22 UTC+1:
>>>> On Tuesday, 22 November 2022 at 15:51:31 UTC-4, WM wrote:
>>>>
>>>> ...nothing
>>> that you could discuss without showing that even your inappropriate limits will not help. A clearer contradiction of your claim is hardly possible:
>>>
>>> If the infinitely many O's leave in the limit, then there is no indexing possible. If they do not leave, then there is no completed indexing.
>> That bullshit only shows that you have no idea how limits work.
>

<snip qUaCk fake math, go play checkers and swaparoo there>

>
> Regards, WM

WM submitted this idea :
> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
>> WM brought next idea :
>>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
>>>
>>>> He has imagined a 'set' of numbers whose elements are included in the
>>>> set by the criterion that they have no criterion -- and thus are dark.
>>>>
>>>> Genius, pure and simple.
>>>>
>>>> Too bad they are useless.
>>>
>>> They are not useless but useful to prove that the idea of equivalent
>>> infinite stes is wrong.
>> Of course it is wrong, the naturals with zero and the naturals without
>> zero are not the same set, they are the same size though.
>
> Two sets are said to be equivalent if they have the same cardinality.

Two sets are said to be the same size (equinumerous) if they have the
same cardinality.

Two sets are said to be equal if each can be a subset of the other.

On 11/23/2022 10:04 AM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
>> WM brought next idea :
>>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
>>>
>>>> He has imagined a 'set' of numbers whose elements are included in the
>>>> set by the criterion that they have no criterion -- and thus are dark.
>>>>
>>>> Genius, pure and simple.
>>>>
>>>> Too bad they are useless.
>>>
>>> They are not useless but useful to prove that the idea of equivalent infinite
>>> stes is wrong.
>> Of course it is wrong, the naturals with zero and the naturals without
>> zero are not the same set, they are the same size though.
>
> Two sets are said to be equivalent if they have the same cardinality.
>
> Regards, WM
>
>

so this {17} is equivalent to this {a} ??

On 11/23/2022 12:52 PM, Sergi o wrote:
> On 11/23/2022 10:04 AM, WM wrote:
>> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
>>> WM brought next idea :
>>>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
>>>>
>>>>> He has imagined a 'set' of numbers whose elements are included in the
>>>>> set by the criterion that they have no criterion -- and thus are dark.
>>>>>
>>>>> Genius, pure and simple.
>>>>>
>>>>> Too bad they are useless.
>>>>
>>>> They are not useless but useful to prove that the idea of equivalent infinite
>>>> stes is wrong.
>>> Of course it is wrong, the naturals with zero and the naturals without
>>> zero are not the same set, they are the same size though.
>>
>> Two sets are said to be equivalent if they have the same cardinality.
>>
>> Regards, WM
>>
>>
>
> so this {17} is equivalent to this {a} ??

{17} is equivalent to

{a} and
{LargeBoat}

so all single member sets are equivalent.

On Wednesday, 23 November 2022 at 12:04:47 UTC-4, WM wrote:
[...]
> Two sets are said to be equivalent if they have the same cardinality.

Two sets are *equinumerous* if they have the same cardinality. But you are so fucking ignorant that you think the set of even naturals and the set of odd naturals are not equinumerous because one of them has one element more than the other (and that neither is equinumerous with the set of naturals). Piss off, wanker.

FromTheRafters schrieb am Mittwoch, 23. November 2022 um 18:16:40 UTC+1:
> WM submitted this idea :
> > FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
> >> WM brought next idea :
> >>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
> >>>
> >>>> He has imagined a 'set' of numbers whose elements are included in the
> >>>> set by the criterion that they have no criterion -- and thus are dark.
> >>>>
> >>>> Genius, pure and simple.
> >>>>
> >>>> Too bad they are useless.
> >>>
> >>> They are not useless but useful to prove that the idea of equivalent
> >>> infinite stes is wrong.
> >> Of course it is wrong, the naturals with zero and the naturals without
> >> zero are not the same set, they are the same size though.
> >
> > Two sets are said to be equivalent if they have the same cardinality.
> Two sets are said to be the same size (equinumerous) if they have the
> same cardinality.
>
> Two sets are said to be equal if each can be a subset of the other.

https://www.google.com/search?q=equivalent+set+example&rlz=1C1CHBF_deDE881DE881&oq=equivalent&aqs=chrome.0.69i59j69i57j0i512l6j0i10i512j0i512.4511j0j7&sourceid=chrome&ie=UTF-8

Regards, WM

Gus Gassmann schrieb am Mittwoch, 23. November 2022 um 20:13:03 UTC+1:
> On Wednesday, 23 November 2022 at 12:04:47 UTC-4, WM wrote:
> [...]
> > Two sets are said to be equivalent if they have the same cardinality.
> Two sets are *equinumerous* if they have the same cardinality.

https://www.google.com/search?q=equivalent+set+example&rlz=1C1CHBF_deDE881DE881&oq=equivalent&aqs=chrome.0.69i59j69i57j0i512l6j0i10i512j0i512.4511j0j7&sourceid=chrome&ie=UTF-8

Regards, WM

WM laid this down on his screen :
> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 18:16:40 UTC+1:
>> WM submitted this idea :
>>> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
>>>> WM brought next idea :
>>>>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
>>>>>
>>>>>> He has imagined a 'set' of numbers whose elements are included in the
>>>>>> set by the criterion that they have no criterion -- and thus are dark.
>>>>>>
>>>>>> Genius, pure and simple.
>>>>>>
>>>>>> Too bad they are useless.
>>>>>
>>>>> They are not useless but useful to prove that the idea of equivalent
>>>>> infinite stes is wrong.
>>>> Of course it is wrong, the naturals with zero and the naturals without
>>>> zero are not the same set, they are the same size though.
>>>
>>> Two sets are said to be equivalent if they have the same cardinality.
>> Two sets are said to be the same size (equinumerous) if they have the
>> same cardinality.
>>
>> Two sets are said to be equal if each can be a subset of the other.
>
> https://www.google.com/search?q=equivalent+set+example&rlz=1C1CHBF_deDE881DE881&oq=equivalent&aqs=chrome.0.69i59j69i57j0i512l6j0i10i512j0i512.4511j0j7&sourceid=chrome&ie=UTF-8
>
> Regards, WM

https://www.google.com/search?client=firefox-b-1-d&sxsrf=ALiCzsbJg_PCJ759GntKa50JcSkyZg3zAA:1669262523635&q=equivalence+in+math&spell=1&sa=X&ved=2ahUKEwi98oe-98X7AhXyGVkFHWGfCrkQirwEKAB6BAgGEAE&biw=1179&bih=581&dpr=1

Sergi o was thinking very hard :
> On 11/23/2022 12:52 PM, Sergi o wrote:
>> On 11/23/2022 10:04 AM, WM wrote:
>>> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
>>>> WM brought next idea :
>>>>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
>>>>>
>>>>>> He has imagined a 'set' of numbers whose elements are included in the
>>>>>> set by the criterion that they have no criterion -- and thus are dark.
>>>>>>
>>>>>> Genius, pure and simple.
>>>>>>
>>>>>> Too bad they are useless.
>>>>>
>>>>> They are not useless but useful to prove that the idea of equivalent
>>>>> infinite
>>>>> stes is wrong.
>>>> Of course it is wrong, the naturals with zero and the naturals without
>>>> zero are not the same set, they are the same size though.
>>>
>>> Two sets are said to be equivalent if they have the same cardinality.
>>>
>>> Regards, WM
>>>
>>>
>>
>> so this {17} is equivalent to this {a} ??
>
> {17} is equivalent to
>
> {a} and
> {LargeBoat}
>
>
> so all single member sets are equivalent.

All natural numbers can be represented by Frege's system. Any set of
cardinality 5 can represent 5. Even {4,3,5,2,1} is 5 but there is now
the notion that a set like 5 (here represented as {4,3,5,2,1}) contains
itself as an element. Self-references can be fun.

There can be an equivalence relation of cardinalities of differing sets
if there is a bijection.

Are we there yet?

onsdag 23 november 2022 kl. 23:03:35 UTC+1 skrev WM:
> FromTheRafters schrieb am Mittwoch, 23. November 2022 um 18:16:40 UTC+1:
> > WM submitted this idea :
> > > FromTheRafters schrieb am Mittwoch, 23. November 2022 um 13:22:48 UTC+1:
> > >> WM brought next idea :
> > >>> FromTheRafters schrieb am Dienstag, 22. November 2022 um 22:31:33 UTC+1:
> > >>>
> > >>>> He has imagined a 'set' of numbers whose elements are included in the
> > >>>> set by the criterion that they have no criterion -- and thus are dark.
> > >>>>
> > >>>> Genius, pure and simple.
> > >>>>
> > >>>> Too bad they are useless.
> > >>>
> > >>> They are not useless but useful to prove that the idea of equivalent
> > >>> infinite stes is wrong.
> > >> Of course it is wrong, the naturals with zero and the naturals without
> > >> zero are not the same set, they are the same size though.
> > >
> > > Two sets are said to be equivalent if they have the same cardinality.
> > Two sets are said to be the same size (equinumerous) if they have the
> > same cardinality.
> >
> > Two sets are said to be equal if each can be a subset of the other.
> https://www.google.com/search?q=equivalent+set+example&rlz=1C1CHBF_deDE881DE881&oq=equivalent&aqs=chrome.0.69i59j69i57j0i512l6j0i10i512j0i512.4511j0j7&sourceid=chrome&ie=UTF-8
>
> Regards, WM
we know that for any infinite cardinality c that for a finite number n, we have |c^n|=|c|
so |N^2| = |N|
We also have |c-n|=c
we then also have |Q+|=|Nx(N/{0})|=|NxN|=|N^2|=|N|

and by definition of how cardinal arithmetic works, we have a bijection from N to Q+

Now shut the fuck up with your bullshit.

zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:

> we know that for any infinite cardinality c that for a finite number n, we have |c^n|=|c|
> so |N^2| = |N|

And I know that you are wrong.
Here is the simplest proof (1 page) that should even be comprehensible by you:
https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
and here is a simplified version (1 page) of the proof in the OP
https://www.researchgate.net/publication/358404509_Countability_Contradicted
Before arguing show that you have understood at least one of them.

Regards, WM

torsdag 24 november 2022 kl. 09:35:56 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
>
> > we know that for any infinite cardinality c that for a finite number n, we have |c^n|=|c|
> > so |N^2| = |N|
> And I know that you are wrong.

I am not, because it is true for ANY cardinality c and finite number n, that means it is more powerful than your garbage.

> Here is the simplest proof (1 page) that should even be comprehensible by you:

Which has many times been debunked here

> https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
> and here is a simplified version (1 page) of the proof in the OP

Still garbage

> https://www.researchgate.net/publication/358404509_Countability_Contradicted
> Before arguing show that you have understood at least one of them.

Why? I have poitned out time and time again that your concept of "dark" and "definable" are MEANINGLESS! They have no meaning in mathematics! You cannot even fucking understand basic, well ANYTHING!

>
> Regards, WM

torsdag 24 november 2022 kl. 09:35:56 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
>
> > we know that for any infinite cardinality c that for a finite number n, we have |c^n|=|c|
> > so |N^2| = |N|
> And I know that you are wrong.
> Here is the simplest proof (1 page) that should even be comprehensible by you:
> https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
> and here is a simplified version (1 page) of the proof in the OP
> https://www.researchgate.net/publication/358404509_Countability_Contradicted
> Before arguing show that you have understood at least one of them.
>
> Regards, WM
https://www.math.ksu.edu/~nagy/real-an/ap-b-card.pdf
Educate your fucking self

On 11/24/2022 2:35 AM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
>
>> we know that for any infinite cardinality c that for a finite number n, we have |c^n|=|c|
>> so |N^2| = |N|
>
> And I know that you are wrong.
> Here is the simplest proof (1 page) that should even be comprehensible by you:
> https://www.researchgate.net/publication/336220780_Dark_natural_numbers_in_set_theory
> and here is a simplified version (1 page) of the proof in the OP
> https://www.researchgate.net/publication/358404509_Countability_Contradicted
> Before arguing show that you have understood at least one of them.
>
> Regards, WM
>

|N^2| = |N| is well known.

above links to WMs papers are not peer reviewed, but self published from WMs postings in sci.math, and still wrong.

On Thursday, November 24, 2022 at 3:17:28 PM UTC+1, Sergi o wrote:
> On 11/24/2022 2:35 AM, WM wrote:
> > zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
> > >
> > > |N^2| = |N|
> > >
> > And I know that you are wrong.
> >
> |N^2| = |N| is well known.

See: https://en.wikipedia.org/wiki/Pairing_function

On 11/24/2022 9:21 AM, Fritz Feldhase wrote:
> On Thursday, November 24, 2022 at 3:17:28 PM UTC+1, Sergi o wrote:
>> On 11/24/2022 2:35 AM, WM wrote:
>>> zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
>>>>
>>>> |N^2| = |N|
>>>>
>>> And I know that you are wrong.
>>>
>> |N^2| = |N| is well known.
>
> See: https://en.wikipedia.org/wiki/Pairing_function

which has

Hopcroft and Ullman pairing function

Cantor pairing function

P2(x,y):=2^{x}* (2y+1)-1

https://www.sciencedirect.com/science/article/pii/002200009290027G?via%3Dihub

torsdag 24 november 2022 kl. 16:21:11 UTC+1 skrev Fritz Feldhase:
> On Thursday, November 24, 2022 at 3:17:28 PM UTC+1, Sergi o wrote:
> > On 11/24/2022 2:35 AM, WM wrote:
> > > zelos...@gmail.com schrieb am Donnerstag, 24. November 2022 um 07:17:14 UTC+1:
> > > >
> > > > |N^2| = |N|
> > > >
> > > And I know that you are wrong.
> > >
> > |N^2| = |N| is well known.
> See: https://en.wikipedia.org/wiki/Pairing_function
not only that

we have injections f:N -> NxN, f(x)=(x,1) and g:NxN->N, g(x,y)=2^x*3^y, which means that a bijection exists

On 11/22/2022 6:03 PM, Sergi o wrote:
> On 11/22/2022 4:35 PM, Chris M. Thomasson wrote:
>> On 11/22/2022 2:11 PM, WM wrote:
>>> Gus Gassmann schrieb am Dienstag, 22. November 2022 um 22:55:22 UTC+1:
>>>> On Tuesday, 22 November 2022 at 15:51:31 UTC-4, WM wrote:
>>>>
>>>> ...nothing
>>>
>>> that you could discuss without showing that even your inappropriate
>>> limits will not help. A clearer contradiction of your claim is hardly
>>> possible:
>>>
>>> If the infinitely many O's leave in the limit, then there is no
>>> indexing possible. If they do not leave, then there is no completed
>>> indexing.
>>
>> Any natural number, yes including zero, creates a unique pair wrt
>> Cantor Pairing. Nothing to miss. Nothing is dark.
>>
>
>
>
> forget dark numbers,  what about dark letters ?
>
>
>
>
> I make e and a dark in above......
>
>     forg█t d█rk numb█rs, wh█t █bout d█rk l█tt█rs ?
>
>
>
>
> here is a dark letter for cut and paste  █   very useful too!

l l ;^)