On 2/23/2022 10:33 AM, FredJeffries wrote:
> On Wednesday, February 23, 2022 at 7:24:21 AM UTC-8, Fritz Feldhase wrote:
>> On Wednesday, February 23, 2022 at 3:38:31 PM UTC+1, horand....@gmail.com wrote:
>>> On Wednesday, 23 February 2022 at 02:04:09 UTC-4, zelos...@gmail.com wrote:
>>>> tisdag 22 februari 2022 kl. 11:42:52 UTC+1 skrev WM:
>>>>>
>>>>> The only escape is to claim that the *limit* is the empty matrix
>>>>>
>>>>> 1/1, __, __, __, ...
>>>>> 1/2, __, __, __, ...
>>>>> 2/1, __, __, __, ...
>>>>> 1/3, __, __, __, ...
>>>>> 2/2, __, __, __, ...
>>>>> :
>>>>>
>>>> Your empty "matrix" is not a matrix, it isn't a function, it is not a mathematical object!
>>>>
>>> That is not quite correct.
>> I think that is ABSOLUTELY correct. Please note that Mückenheim's "empty matrix"
>>> 1/1, __, __, __, ...
>>> 1/2, __, __, __, ...
>>> 2/1, __, __, __, ...
>>> 1/3, __, __, __, ...
>>> 2/2, __, __, __, ...
>>> :
>>
>> is not identical with your matrix:
>>> 1/1, 0, 0, 0, ...
>>> 1/2, 0, 0, 0, ...
>>> 2/1, 0, 0, 0, ...
>>> 1/3, 0, 0, 0, ...
>>> 2/2, 0, 0, 0, ...
>>> :
>>
>> You see the difference?
>
> That is a good point. Perhaps it might be better to consider the datatype of the 'matrix' entries to be 'nullable fractions' rather than fractions.
>
> https://en.wikipedia.org/wiki/Nullable_type

WM's math is both "self-nulling" and "impotentially exists"

onsdag 23 februari 2022 kl. 12:46:53 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 23. Februar 2022 um 07:04:09 UTC+1:
> > tisdag 22 februari 2022 kl. 11:42:52 UTC+1 skrev WM:
> > > horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 18:35:39 UTC+1:
> > > > On Monday, 21 February 2022 at 13:20:52 UTC-4, WM wrote:
> > > > > horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
> > >
> > > > > Counting means that every fraction gets a natnumber. If never an empty place occurs but in the limit there are all places empty, then infinitely many fractions have not been counted.
> > > > Borrring! Of course for every natural number k there are aleph_0 natural numbers that are larger than k.
> > > That proves that Cantor's claim of counting all is nonsense. But it can nowhere be seen as clear as here:
> > >
> > > For *every* fraction
> > > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > > that Cantor counts by
> > > k = (m + n - 1)(m + n - 2)/2 + m
> > > there is a matrix related to this very fraction, because this fraction is in its first column.
> > >
> > > 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
> > > 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
> > > 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> > > ... ... ... ...
> > >
> > > And this matrix is completely filled. The only escape is to claim that the *limit* is the empty matrix
> > >
> > > 1/1, __, __, __, ...
> > > 1/2, __, __, __, ...
> > > 2/1, __, __, __, ...
> > > 1/3, __, __, __, ...
> > > 2/2, __, __, __, ...
> > > ...
> > >
> > > But here nobody will believe that between all natural numbers and the limit all columns can get empty. That would require infinitely many steps or natural numbers between all natural numbers leaving the matrix filled and the limit.
> > >
> > > Dark numbers are the only explanation. Note that every lit fraction will appear in the first column and no lit fraction can be found fixed elsewhere in the completely filled matrix.
> > >
> > your empty "matrix" is not a matrix, it isn't a function,
> In fact, it will never be empty. But according to Cantor it should be this matrix:

It isn't anything, it is not a mathematical object!

> 1/1, __, __, __, ...
> 1/2, __, __, __, ...
> 2/1, __, __, __, ...
> 1/3, __, __, __, ...
> 2/2, __, __, __, ...
> ...
> > it is not a mathematical object!
> Why?

Because it is not defined in any meaningful way mathematically!
Matrices cannot have "empty slots" by definition.

>
> Retards, WM

onsdag 23 februari 2022 kl. 15:38:31 UTC+1 skrev horand....@gmail.com:
> On Wednesday, 23 February 2022 at 02:04:09 UTC-4, zelos...@gmail.com wrote:
> > tisdag 22 februari 2022 kl. 11:42:52 UTC+1 skrev WM:
> > > horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 18:35:39 UTC+1:
> > > > On Monday, 21 February 2022 at 13:20:52 UTC-4, WM wrote:
> > > > > horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
> > >
> > > > > Counting means that every fraction gets a natnumber. If never an empty place occurs but in the limit there are all places empty, then infinitely many fractions have not been counted.
> > > > Borrring! Of course for every natural number k there are aleph_0 natural numbers that are larger than k.
> > > That proves that Cantor's claim of counting all is nonsense. But it can nowhere be seen as clear as here:
> > >
> > > For *every* fraction
> > > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > > that Cantor counts by
> > > k = (m + n - 1)(m + n - 2)/2 + m
> > > there is a matrix related to this very fraction, because this fraction is in its first column.
> > >
> > > 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, .... 1/1, 3/1, 4/1, 1/4, ...
> > > 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, .... 1/2, 5/1, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, .... 2/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, .... 1/3, 4/2, 4/3, 4/4, ...
> > > 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, .... 2/2, 5/2, 5/3, 5/4, ...
> > > ... ... ... ...
> > >
> > > And this matrix is completely filled. The only escape is to claim that the *limit* is the empty matrix
> > >
> > > 1/1, __, __, __, ...
> > > 1/2, __, __, __, ...
> > > 2/1, __, __, __, ...
> > > 1/3, __, __, __, ...
> > > 2/2, __, __, __, ...
> > > ...
> > >
> > > But here nobody will believe that between all natural numbers and the limit all columns can get empty. That would require infinitely many steps or natural numbers between all natural numbers leaving the matrix filled and the limit.
> > >
> > > Dark numbers are the only explanation. Note that every lit fraction will appear in the first column and no lit fraction can be found fixed elsewhere in the completely filled matrix.
> > >
> > > Regards, WM
> > your empty "matrix" is not a matrix, it isn't a function, it is not a mathematical object!
> That is not quite correct. He has described a process by which you can associate with each natural number k a matrix M(k), with describable transitions from M(k) to M(k+1). In fact, you can show that the first column will at step k have the first k fractions in the top k slots and other fractions below, and the rest of the columns have entries n/m for which n + m is at least (k^2)/2. As k-> oo, the matrices M(k) will indeed converge in a weird sense (elementwise to the fraction enumerated by the Cantor enumeration in the first column, and elementwise to 0 in the other columns). This limit is
>
> 1/1, 0, 0, 0, ...
> 1/2, 0, 0, 0, ...
> 2/1, 0, 0, 0, ...
> 1/3, 0, 0, 0, ...
> 2/2, 0, 0, 0, ...
> 3/1, 0, 0, 0, ...
> ...
>
> In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.

But that is not what he argues and the refutation of that one is because the initial matrix does not contain infinite 0s, hence it cannot be made into that.

horand....@gmail.com schrieb am Mittwoch, 23. Februar 2022 um 15:38:31 UTC+1:

> > your empty "matrix" is not a matrix, it isn't a function, it is not a mathematical object!
> That is not quite correct. He has described a process by which you can associate with each natural number k a matrix M(k), with describable transitions from M(k) to M(k+1). In fact, you can show that the first column will at step k have the first k fractions in the top k slots and other fractions below, and the rest of the columns have entries n/m for which n + m is at least (k^2)/2. As k-> oo, the matrices M(k) will indeed converge in a weird sense (elementwise to the fraction enumerated by the Cantor enumeration in the first column, and elementwise to 0 in the other columns). This limit is
>
> 1/1, 0, 0, 0, ...
> 1/2, 0, 0, 0, ...
> 2/1, 0, 0, 0, ...
> 1/3, 0, 0, 0, ...
> 2/2, 0, 0, 0, ...
> 3/1, 0, 0, 0, ...
> ...
>
> In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.

There are no zeros coming up. Never a place without a positive fraction occurs.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 23. Februar 2022 um 16:18:03 UTC+1:

> Can you actually PROVE your claim? (I got the impression that actually, say, (M_k)_1,2 --> oo. So far I see (M_k)_1,2 = (2/1, 3/1, 6/1, ...) and a little consideration shows that (M_k)_1,2 diverges, no?)

Like the sequence 1, 2, 3, ... . Important is only that never an empty place can occur when two fractions are exchanged.

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:12:56 UTC+1:
> onsdag 23 februari 2022 kl. 12:46:53 UTC+1 skrev WM:

> > > it is not a mathematical object!
> > Why?
> Because it is not defined in any meaningful way mathematically!
> Matrices cannot have "empty slots" by definition.

My matrices have no empty slots by mathematical proof: When two positive fractions change their places they do not leave an empty place. Thus your statement is wrong.

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:13:48 UTC+1:
> onsdag 23 februari 2022 kl. 15:38:31 UTC+1 skrev horand....@gmail.com:

> > This limit is
> >
> > 1/1, 0, 0, 0, ...
> > 1/2, 0, 0, 0, ...
> > 2/1, 0, 0, 0, ...
> > 1/3, 0, 0, 0, ...
> > 2/2, 0, 0, 0, ...
> > 3/1, 0, 0, 0, ...
> > ...
> >
> > In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.
> But that is not what he argues and the refutation of that one is because the initial matrix does not contain infinite 0s, hence it cannot be made into that.

So it is. Exchanging places leaves all fractions within the matrix, only all definable fractions
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
defined by Cantor's
k = (m + n - 1)(m + n - 2)/2 + m
are gathered in the first column.

There can be no doubt. Each fraction is in the first column and can be found there, and none is in the remaining columns permanently. That is fact. Mathematical fact. But fact is also, that never an empty place occurs.

Regards, WM

On 2/24/2022 6:59 AM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:12:56 UTC+1:
>> onsdag 23 februari 2022 kl. 12:46:53 UTC+1 skrev WM:
>
>>>> it is not a mathematical object!
>>> Why?
>> Because it is not defined in any meaningful way mathematically!
>> Matrices cannot have "empty slots" by definition.
>
> My matrices have no empty slots by mathematical spoof: When two positive fractions change their places they do not leave an empty place. Thus your statement is wrong.
>
> Regards, WM

then why do you show a matrix with "empty slots" ?

On 2/24/2022 6:54 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 23. Februar 2022 um 16:18:03 UTC+1:
>
>> Can you actually PROVE your claim? (I got the impression that actually, say, (M_k)_1,2 --> oo. So far I see (M_k)_1,2 = (2/1, 3/1, 6/1, ...) and a little consideration shows that (M_k)_1,2 diverges, no?)
>
> Like the sequence 1, 2, 3, ... . Important is only that never an empty place can occur when two fractions are exchanged.
>
> Regards, WM

Diversion. WM did not understand your question.

On Thursday, February 24, 2022 at 2:44:32 PM UTC+1, horand....@gmail.com wrote:

> [...] His claim that there will always be positive fractions, even in the limit, is, of course, nonsense.

Espescially since THERE IS NO limit (of the sequence of matrices).

On the other hand, it's a simple fact that each and every matrix in the sequence is "completely filled" with fractions.

> So one has to argue a bit more carefully. There is no doubt that the fractions n/m in places other than in the first column of M(k) will have larger and larger sums n+m as k -> oo. There will be some elements that converge to 0, others that converge to oo, and I conjecture that the rest (if there are such) will oscillate, with the two accumulation points being 0 and oo. At any rate, every positive fraction that is not originally in the first column will eventually be swapped there, and after that it will remain in situ. What happens to the rest of the matrix is then irrelevant.

But we are not interested in the outcome of a "supertask" (since this does not count as a mathematical argument in set theory) nor in the lim of the sequence of marices (which does not exist).

So why consider that nonsensical approach AT ALL?

On Thursday, February 24, 2022 at 2:56:46 PM UTC+1, sergio wrote:

> then why do you show a matrix with "empty slots" ?

Because he is a complete idiot (and/or a con-artist).

On 2/24/2022 6:51 AM, WM wrote:
> horand....@gmail.com schrieb am Mittwoch, 23. Februar 2022 um 15:38:31 UTC+1:
>
>>> your empty "matrix" is not a matrix, it isn't a function, it is not a mathematical object!
>> That is not quite correct. He has described a process by which you can associate with each natural number k a matrix M(k), with describable transitions from M(k) to M(k+1). In fact, you can show that the first column will at step k have the first k fractions in the top k slots and other fractions below, and the rest of the columns have entries n/m for which n + m is at least (k^2)/2. As k-> oo, the matrices M(k) will indeed converge in a weird sense (elementwise to the fraction enumerated by the Cantor enumeration in the first column, and elementwise to 0 in the other columns). This limit is
>>
>> 1/1, 0, 0, 0, ...
>> 1/2, 0, 0, 0, ...
>> 2/1, 0, 0, 0, ...
>> 1/3, 0, 0, 0, ...
>> 2/2, 0, 0, 0, ...
>> 3/1, 0, 0, 0, ...
>> ...
>>
>> In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.
>
> There are no zeros coming up. Never a place without a positive fraction occurs.
>
> Regards, WM

no need for all that, from the matrix of rationals, Cantor converted it to a linear array, and that is almost what you did here, as your first column is
the only thing needed, the rest of the matrix is bogus, but you dont know algebra, so this is what you do.

On Thursday, February 24, 2022 at 3:01:17 PM UTC+1, sergio wrote:
> On 2/24/2022 6:54 AM, WM wrote:
> > Fritz Feldhase schrieb am Mittwoch, 23. Februar 2022 um 16:18:03 UTC+1:
> > >
> > > I got the impression that actually, say, (M_k)_1,2 --> oo. So far I see ((M_k)_1,2) = (2/1, 3/1, 6/1, ...) and a little consideration shows that ((M_k)_1,2) diverges, no?
> > >
> > Like the sequence 1, 2, 3, ... . Important is only that never an empty place can occur when two fractions are exchanged.
> >
> [...] WM did not understand your question.

Right. If the sequence were, say, (1/1, 1/2, 1/3, 1/4, ...) it would _converge_.

Has nothing to do with the fact that "never an empty place can occur when two fractions are exchanged". (*sigh*)

On 2/24/2022 7:06 AM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:13:48 UTC+1:
>> onsdag 23 februari 2022 kl. 15:38:31 UTC+1 skrev horand....@gmail.com:
>
>>> This limit is
>>>
>>> 1/1, 0, 0, 0, ...
>>> 1/2, 0, 0, 0, ...
>>> 2/1, 0, 0, 0, ...
>>> 1/3, 0, 0, 0, ...
>>> 2/2, 0, 0, 0, ...
>>> 3/1, 0, 0, 0, ...
>>> ...
>>>
>>> In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.
>> But that is not what he argues and the refutation of that one is because the initial matrix does not contain infinite 0s, hence it cannot be made into that.
>
> So it is. Exchanging places leaves all fractions within the matrix, only all definable fractions
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> defined by Cantor's
> k = (m + n - 1)(m + n - 2)/2 + m
> are gathered in the first column.
>
> There can be no doubt. Each fraction is in the first column and can be found there, and none is in the remaining columns permanently. That is fact. Mathematical fact. But fact is also, that never an empty place occurs.
>
> Regards, WM

Wrong. Where did all those 0's filling up the rest of the matrix come from ?

WM laid this down on his screen :
> Fritz Feldhase schrieb am Mittwoch, 23. Februar 2022 um 16:18:03 UTC+1:
>
>> Can you actually PROVE your claim? (I got the impression that actually, say,
>> (M_k)_1,2 --> oo. So far I see (M_k)_1,2 = (2/1, 3/1, 6/1, ...) and a little
>> consideration shows that (M_k)_1,2 diverges, no?)
>
> Like the sequence 1, 2, 3, ... . Important is only that never an empty place
> can occur when two fractions are exchanged.

How many permutations are there of the natural numbers? Can NxN hold
them all?

FromTheRafters schrieb am Donnerstag, 24. Februar 2022 um 16:02:37 UTC+1:

> How many permutations are there of the natural numbers? Can NxN hold
> them all?

We need only the exchanges of fractions of Cantor's sequence

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

and the original inhabitants of their places

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

Regards, WM

horand....@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 14:44:32 UTC+1:
> On Thursday, 24 February 2022 at 02:13:48 UTC-4, zelos...@gmail.com wrote:
> His claim that there will always be positive fractions, even in the limit, is, of course, nonsense.

Why should it? Because set theorists are too stupid to understaned that be exchanging fractions never an empty position can appear?

> So one has to argue a bit more carefully.

> At any rate, every positive fraction that is not originally in the first column will eventually be swapped there, and after that it will remain in situ. What happens to the rest of the matrix is then irrelevant.

That is very careless. Fact is: If all fractions are gathered in a zeroth column or elsewehre, then we get the same sequenc as when they are gathered in the first column. The only difference is this: When gathering them in the zeroth column you are no aware that almost all fractions remain in the matrix because they are dark. When gathering them in the first column, this cannot be overlooked.

Regards, WM

sergio schrieb am Donnerstag, 24. Februar 2022 um 14:56:46 UTC+1:
> On 2/24/2022 6:59 AM, WM wrote:
> > zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:12:56 UTC+1:
> >> onsdag 23 februari 2022 kl. 12:46:53 UTC+1 skrev WM:
> >
> >>>> it is not a mathematical object!
> >>> Why?
> >> Because it is not defined in any meaningful way mathematically!
> >> Matrices cannot have "empty slots" by definition.
> >
> > My matrices have no empty slots by mathematical proof: When two positive fractions change their places they do not leave an empty place. Thus your statement is wrong.

> then why do you show a matrix with "empty slots" ?

Because that is what should be the result of the process according to Cantor and his disciples.

Regards, WM

Fritz Feldhase schrieb am Donnerstag, 24. Februar 2022 um 15:01:26 UTC+1:
> On Thursday, February 24, 2022 at 2:44:32 PM UTC+1, horand....@gmail.com wrote:
>
> > [...] His claim that there will always be positive fractions, even in the limit, is, of course, nonsense.
>
> Espescially since THERE IS NO limit (of the sequence of matrices).

And there is no limit of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>
> On the other hand, it's a simple fact that each and every matrix in the sequence is "completely filled" with fractions.

> But we are not interested in the outcome of a "supertask" (since this does not count as a mathematical argument in set theory) nor in the lim of the sequence of marices (which does not exist).

This sequence is not a super task but as static as Cantor's sequence

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

Regards, WM

horand....@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 16:00:27 UTC+1:

> Well, I happen to think that it is an interesting question how the individual elements m_{i,j}(k) of the matrices M(k) behave as k -> oo.

They do not "behave" at all. The matrix is filled with positive fractions which, whether being shuffled or not, will remain positive fractions in the matrix.

> It is clearly not a matrix limit in any traditional sense

like 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ... has no limit.

> like it or not, the sequence of matrices is out there,

In fact, so it is.

> Better to have a counter other than that he understands nothing

That will remain your best defence. How can you have another counter if you understand nothing yourself?

Regards, WM

On 2/24/2022 8:11 AM, Fritz Feldhase wrote:
> On Thursday, February 24, 2022 at 3:01:17 PM UTC+1, sergio wrote:
>> On 2/24/2022 6:54 AM, WM wrote:
>>> Fritz Feldhase schrieb am Mittwoch, 23. Februar 2022 um 16:18:03 UTC+1:
>>>>
>>>> I got the impression that actually, say, (M_k)_1,2 --> oo. So far I see ((M_k)_1,2) = (2/1, 3/1, 6/1, ...) and a little consideration shows that ((M_k)_1,2) diverges, no?
>>>>
>>> Like the sequence 1, 2, 3, ... . Important is only that never an empty place can occur when two fractions are exchanged.
>>>
>> [...] WM did not understand your question.
>
> Right. If the sequence were, say, (1/1, 1/2, 1/3, 1/4, ...) it would _converge_.
>
> Has nothing to do with the fact that "never an empty place can occur when two fractions are exchanged". (*sigh*)

like trying to help a guy lost in the woods, then he goes even more lost-er into the woods.

On 2/24/2022 9:42 AM, WM wrote:
> FromTheRafters schrieb am Donnerstag, 24. Februar 2022 um 16:02:37 UTC+1:
>
>> How many permutations are there of the natural numbers? Can NxN hold
>> them all?
>
> We need only the exchanges of fractions of Cantor's sequence
>
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>
> and the original inhabitants of their places
>
> 1/1, 2/1, 1/3, 1/4, ... 1/1, 3/1, 1/3, 1/4, ... 1/1, 3/1, 4/1, 1/4, ... 1/1, 3/1, 4/1, 1/4, ...
> 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 2/2, 2/3, 2/4, ... 1/2, 5/1, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ... 2/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ... 4/1, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ... 1/3, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> ... ... ... ...
>
> Regards, WM

no inhabitants of matrices are needed, no transpositions are needed.
Only an understanding of algebra is needed.

On 2/24/2022 9:52 AM, WM wrote:
> sergio schrieb am Donnerstag, 24. Februar 2022 um 14:56:46 UTC+1:
>> On 2/24/2022 6:59 AM, WM wrote:
>>> zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:12:56 UTC+1:
>>>> onsdag 23 februari 2022 kl. 12:46:53 UTC+1 skrev WM:
>>>
>>>>>> it is not a mathematical object!
>>>>> Why?
>>>> Because it is not defined in any meaningful way mathematically!
>>>> Matrices cannot have "empty slots" by definition.
>>>
>>> My matrices have no empty slots by mathematical proof: When two positive fractions change their places they do not leave an empty place. Thus your statement is wrong.
>
>> then why do you show a matrix with "empty slots" ?
>
> Because that is what should be the result of the process according to Cantor and his disciples.
>
> Regards, WM

Don't blame your "math" failures on Cantor.
Now man-up and admit your approach has no way of ever working.

On 2/24/2022 8:12 AM, sergio wrote:
> On 2/24/2022 7:06 AM, WM wrote:
>> zelos...@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 07:13:48 UTC+1:
>>> onsdag 23 februari 2022 kl. 15:38:31 UTC+1 skrev horand....@gmail.com:
>>
>>>> This limit is
>>>>
>>>> 1/1, 0, 0, 0, ...
>>>> 1/2, 0, 0, 0, ...
>>>> 2/1, 0, 0, 0, ...
>>>> 1/3, 0, 0, 0, ...
>>>> 2/2, 0, 0, 0, ...
>>>> 3/1, 0, 0, 0, ...
>>>> ...
>>>>
>>>> In this sense the elements to the right indeed are not empty, but they are no longer positive fractions, either.
>>> But that is not what he argues and the refutation of that one is because the initial matrix does not contain infinite 0s, hence it cannot be made
>>> into that.
>>
>> So it is. Exchanging places leaves all fractions within the matrix, only all definable fractions
>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>> defined by Cantor's
>> k = (m + n - 1)(m + n - 2)/2 + m
>> are gathered in the first column.

that is no good, as 1/10 is overcounted as 1/10 = 10/100 = 100/1000 = 1000/10000

>>
>> There can be no doubt. Each fraction is in the first column and can be found there, and  none is in the remaining columns permanently. That is fact.
>> Mathematical fact. But fact is also, that never an empty place occurs.
>>
>> Regards, WM
>
> Wrong.  Where did all those 0's filling up the rest of the matrix come from ?

Who is generating the 0's ??

where do they come from ?

On 2/24/2022 9:55 AM, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 24. Februar 2022 um 15:01:26 UTC+1:
>> On Thursday, February 24, 2022 at 2:44:32 PM UTC+1, horand....@gmail.com wrote:
>>
>>> [...] His claim that there will always be positive fractions, even in the limit, is, of course, nonsense.
>>
>> Espescially since THERE IS NO limit (of the sequence of matrices).
>
> And there is no limit of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

that is what the three dots mean ...

> Regards, WM