On 2/21/2022 4:55 AM, Fritz Feldhase wrote:
> On Monday, February 21, 2022 at 9:53:46 AM UTC+1, WM wrote:
>>
>> My proof
>
> Your WHAT?!

Spoof, not proof.

On 2/21/2022 1:24 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 21. Februar 2022 um 06:57:26 UTC+1:
>
>> And logic reveals you are wrong.
>
> Not logic! Every fraction that can be found in Cantor's sequence can be found in the first column. No fraction can be found in the remaining columns. That shows identity of my sequence with Cantor's sequence. But never an empty place can be found either. That shows the existence of dark fractions.
>
> Regards, WM

No. Try again.

WM explained :
> FromTheRafters schrieb am Montag, 21. Februar 2022 um 09:41:09 UTC+1:
>> WM presented the following explanation :
>
>>> Every fraction that can be found in Cantor's sequence can be found in the
>>> first column.
>> Good, because Cantor was counting rational numbers, not fractions.
>
> Wrong. k = (m + n - 1)(m + n - 2)/2 + m counts
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...

Which are indeed rational numbers. Note the countability of the
numerators and the denominators which are all from the set of naturals.

Fritz Feldhase presented the following explanation :
> On Monday, February 21, 2022 at 9:53:46 AM UTC+1, WM wrote:
>>
>> My proof
>
> Your WHAT?!

Poof. Sometimes he mistypes.

FromTheRafters schrieb am Montag, 21. Februar 2022 um 14:35:03 UTC+1:
> WM explained :
> > FromTheRafters schrieb am Montag, 21. Februar 2022 um 09:41:09 UTC+1:
> >> WM presented the following explanation :
> >
> >>> Every fraction that can be found in Cantor's sequence can be found in the
> >>> first column.
> >> Good, because Cantor was counting rational numbers, not fractions.
> >
> > Wrong. k = (m + n - 1)(m + n - 2)/2 + m counts
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> Which are indeed rational numbers.

But not different natural numbers. Cantor counted 1/1 and 2/2 and 3/3 as different.
Had he counted rational numbers, then 1 would have been counted only once.

Simply deny your ingnorance and confusion. The gang of stupid fools of matheology will support you.

Regards, WM

horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
> On Monday, 21 February 2022 at 03:33:48 UTC-4, WM wrote:

> > > > Matheology is based on the contradiciton: If a bijection exists then not every injection must be a bijection.
> > > How is that a contradiction????
> > Because bijection proves equinumerosity, not only equicardinality.
> Please define each of those terms, and explain what you think the difference might be.

Equinumerosity means: If a bijection exists then every injection must be a bijection.
Equicardinality means a bijection between visible elements.

> In particular explain the relationship between the set of positive *even* integers and the set of positive *odd* integers.

All infinite sets have a potentially infinite visible part which has same cardinality. Best this can be proved by the enumeration of the fractions: All visible fractions can be tranferred into the first column and appear there in bijection with the visible natural numbers. The set of all positive fractions has many more numbers, namely |Q+| = |ℕ|^2.

Regards, WM

WM formulated on Monday :
> FromTheRafters schrieb am Montag, 21. Februar 2022 um 14:35:03 UTC+1:
>> WM explained :
>>> FromTheRafters schrieb am Montag, 21. Februar 2022 um 09:41:09 UTC+1:
>>>> WM presented the following explanation :
>>>>> Every fraction that can be found in Cantor's sequence can be found in the
>>>>> first column.
>>>> Good, because Cantor was counting rational numbers, not fractions.
>>>
>>> Wrong. k = (m + n - 1)(m + n - 2)/2 + m counts
>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>> Which are indeed rational numbers.
>
> But not different natural numbers. Cantor counted 1/1 and 2/2 and 3/3 as
> different.

Different members of the same equivalence class.

> Had he counted rational numbers, then 1 would have been counted
> only once.

It doesn't matter, they're countable either way.

> Simply deny your ingnorance and confusion.

Mine?

> The gang of stupid fools of matheology will support you.

I don't need your support.

On 2/21/2022 9:27 AM, WM wrote:
> FromTheRafters schrieb am Montag, 21. Februar 2022 um 14:35:03 UTC+1:
>> WM explained :
>>> FromTheRafters schrieb am Montag, 21. Februar 2022 um 09:41:09 UTC+1:
>>>> WM presented the following explanation :
>>>
>>>>> Every fraction that can be found in Cantor's sequence can be found in the
>>>>> first column.
>>>> Good, because Cantor was counting rational numbers, not fractions.
>>>
>>> Wrong. k = (m + n - 1)(m + n - 2)/2 + m counts
>>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>> Which are indeed rational numbers.
>
> But not different natural numbers. Cantor counted 1/1 and 2/2 and 3/3 as different.

as he should have, each of those is indexed with a different natural number, as they are different.

> Had he counted rational numbers, then 1 would have been counted only once.

No. You are assuming reduced form, that obviously does not apply here.

Would 10/100 not be counted as it is 1/10 in reduced form ?

Its amusing to watch you thrash about inside your world of illogical reasoning.

>
> Simply deny your ingnorance and confusion.

you are projecting again.

> Regards, WM

On 2/21/2022 9:36 AM, WM wrote:
> horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
>> On Monday, 21 February 2022 at 03:33:48 UTC-4, WM wrote:
>
>>>>> Matheology is based on the contradiciton: If a bijection exists then not every injection must be a bijection.
>>>> How is that a contradiction????
>>> Because bijection proves equinumerosity, not only equicardinality.
>> Please define each of those terms, and explain what you think the difference might be.
>
> Equinumerosity means: If a bijection exists then every injection must be a bijection.

No. two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a
function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y.[1]

> Equicardinality means a bijection between visible elements.

No. "visible" requires external observers.

>
>> In particular explain the relationship between the set of positive *even* integers and the set of positive *odd* integers.
>

<snip made up crap>

>
> Regards, WM

FromTheRafters schrieb am Montag, 21. Februar 2022 um 16:41:37 UTC+1:
> WM formulated on Monday :

> >>>> Good, because Cantor was counting rational numbers, not fractions.
> >>>
> >Cantor counted 1/1 and 2/2 and 3/3 as different.

> > Had he counted rational numbers, then 1 would have been counted
> > only once.
> It doesn't matter, they're countable either way.

Therefore your above statement is not only wrong but irrelevant.

Regards, WM

horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
> On Monday, 21 February 2022 at 03:33:48 UTC-4, WM wrote:

> > If never an empty place occurs then the limit has no empty place(s).
> Claims the great (falsely and without a shred of evidence).

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.

Regards, WM

On 2/21/2022 11:12 AM, WM wrote:
> FromTheRafters schrieb am Montag, 21. Februar 2022 um 16:41:37 UTC+1:
>> WM formulated on Monday :
>
>>>>>> Good, because Cantor was counting rational numbers, not fractions.
>>>>>
>>> Cantor counted 1/1 and 2/2 and 3/3 as different.
>
>>> Had he counted rational numbers, then 1 would have been counted
>>> only once.
>> It doesn't matter, they're countable either way.
>
> Therefore your above statement is not only wrong but irrelevant.
>
> Regards, WM

and Cantor counted 1/10, 10/100, 100/1000, .... which are all the same reduced number

another red herring from WM the misdirectionator.

On 2/21/2022 11:20 AM, WM wrote:
> horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 16:10:21 UTC+1:
>> On Monday, 21 February 2022 at 03:33:48 UTC-4, WM wrote:
>
>>> If never an empty place occurs then the limit has no empty place(s).
>> Claims the great (falsely and without a shred of evidence).
>
> Counting means that every fraction gets a natnumber.

No, that is enumeration.

> If never an empty place occurs but in the limit there are all places empty, then infinitely many fractions have not been counted.

all are counted.

Your tic-tac-toe puzzle is diversion.

>
> Regards, WM

why do you persist in this obviously goofy always fail construct ?

It happens that WM formulated :
> FromTheRafters schrieb am Montag, 21. Februar 2022 um 16:41:37 UTC+1:
>> WM formulated on Monday :
>
>>>>>> Good, because Cantor was counting rational numbers, not fractions.
>>>>>
>>> Cantor counted 1/1 and 2/2 and 3/3 as different.
>
>>> Had he counted rational numbers, then 1 would have been counted
>>> only once.
>> It doesn't matter, they're countable either way.
>
> Therefore your above statement is not only wrong but irrelevant.

Neither wrong nor irrelevant. Your failure to see this is telling.

måndag 21 februari 2022 kl. 08:24:29 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Montag, 21. Februar 2022 um 06:57:26 UTC+1:
>
> > And logic reveals you are wrong.
> Not logic!

Yes, logic reveals you are 100% wrong always.

>Every fraction that can be found in Cantor's sequence can be found in the first column.

Your column is not a function or anything meaningful. It is only to obfuscate.

>No fraction can be found in the remaining columns. That shows identity of my sequence with Cantor's sequence. But never an empty place can be found either. That shows the existence of dark fractions.

Shows nothing of the sort. It only shows that your method of shit is just that, shit and garbage.

>
> Regards, WM

horand....@gmail.com schrieb am Montag, 21. Februar 2022 um 18:30:51 UTC+1:
> On Monday, 21 February 2022 at 11:36:39 UTC-4, WM wrote:

> > All infinite sets have a potentially infinite visible part which has same cardinality. Best this can be proved by the enumeration of the fractions: All visible fractions can be tranferred into the first column and appear there in bijection with the visible natural numbers. The set of all positive fractions has many more numbers, namely |Q+| = |ℕ|^2.
> I did not ask about N and Q+. I asked specifically about the odd and even natural numbers. Are they equinumerous?

Yes, at least up to +-1.

> Do they have the same cardinality? What?

Every infinite set M has cardinality ℵo because all finite initial segments can be put in bijection with intial segments of other infinite sets, and the remainder is called ℵo which is nothing but a synonym for actual infinity.
ℵo + n = ℵo. ℵo + |M| = ℵo.

Regards, 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

On Tuesday, February 22, 2022 at 11:42:52 AM UTC+1, WM wrote:

> Dark numbers are the only explanation.

Sure.

Now please fuck off, asshole!

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!

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:

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

> it is not a mathematical object!

Why?

Retards, WM

On 2/23/2022 5:46 AM, WM wrote:

<snip crap>

>>>
>> your empty "matrix" is not a matrix, it isn't a function,
>
> In fact, it will never be empty. But according to Cantor

Liar.

>
> Retards, WM

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:
> >
> > your empty "matrix" is not a matrix, it isn't a function, it is not a mathematical object!

Agree.

> That is not quite correct.

Well... What are the arguments or your claim?

> 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).

Right. Hence we have a sequence (M_i) (with M_i = M(i)) of matrices.

> In fact, you can show that the first column will at step k have the first k fractions [of the sequence of fractions defined by Cantor --FF] in the top k slots and other fractions below,

Agree (of course).

> and the rest of the columns have [some other --FF] entries n/m [...]

> 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, ...
> ...

I don't think that a sensible defintion of "limit" for a sequence of matrices will deliver this result. (But maybe I'm wrong.)

So let's agree that

lim_(k->oo) M_k = R with R_i,i = lim_(k->oo) (M_k)_i,j) if the latter lim exists for all i,k.

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?)

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?

Defending Mückenheim's idiotic claims? Really?

On Wednesday, February 23, 2022 at 4:18:03 PM UTC+1, Fritz Feldhase wrote:
> On Wednesday, February 23, 2022 at 3:38:31 PM UTC+1, horand....@gmail.com wrote:
> >
> > 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, ...
> > ...
> I don't think that a sensible defintion of "limit" for a sequence of matrices will deliver this result. (But maybe I'm wrong.)
>
> So let's agree that
>
> lim_(k->oo) M_k = R with R_i,i = lim_(k->oo) (M_k)_i,j if the latter lim exists for all i,k.

Correction ("improved" the notation):

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?

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