On 2/24/2022 9:50 AM, WM wrote:
> 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?

no, exchanging fractions is stupid.

>
>> 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.

your argument is moot.

Cantor's enumeration of the rationals has already happened, long before your transmuted matrices infected themselves.

>
> Regards, WM

On Thursday, 24 February 2022 at 17:16:43 UTC-4, sergio wrote:
> 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 ...

Well, sort of. The sequence of fractions enumerated by the Cantor function does not have a limit. All you can say is that his interjection was inapt and misdirecting.

torsdag 24 februari 2022 kl. 13:59:32 UTC+1 skrev 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
The one you want as "final" is not a matrix.

And your matrix sequence is irrelevant to the function from N to Q+

On 2/24/2022 6:02 AM, Fritz Feldhase wrote:
> 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).

A con-artist that is adept at fooling hardcore morons? I wonder what he
thinks of his students?

On 2/24/2022 4:54 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.

1, 2, 3, ... can be used to index every fraction.

On 2/24/2022 11:07 AM, sergio wrote:
> 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.

Oh... Uhhh, are these tracks mine? Does a circle exist? wow.

horand....@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 23:25:34 UTC+1:
> I am not interested in the sequence of fractions; I am interested in the individual elements of the matrices M(k).
>
> 1/1, 1/2, 1/3, 1/4, ... 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, 6/1, 4/1, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ... 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, ... 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, ... 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, ... 5/1, 5/2, 5/3, 5/4, ... 2/2, 5/2, 5/3, 5/4, ...
> 6/1, 6/2, 6/3, 6/4, ... 6/1, 6/2, 6/3, 6/4, ... 6/1, 6/2, 6/3, 6/4, ... 6/1, 6/2, 6/3, 6/4, ... 3/1, 6/2, 6/3, 6/4, ...
> ... ... ... ...
>
> The elements in the first column will, of course, be the positive fractions in Cantor's order, and after step k, m_{k,1} will be constant, since the k'th fraction will have been swapped into place. But look at m_{1,2}(k), for instance. The sequence starts 1/2, 2/1, 3/1, 3/1, 6/1, .... How does it continue, and does it have a limit? If it does have a limit, do you think it is one of the fractions you swapped into the first column? (Which one?)

It has no limit. It vanishes in darkness. Note that only fractions which have been there in the matrix are changing places. There were no "limits" in the first matrix and there are no limits later. But there are only potentially many lit fractions. Every place of the matrix has its sequence. The places in the first column get settled. The other places get different sequences of fractions with no end. You can extend 1/2, 2/1, 3/1, 3/1, 6/1, ... as far as you like. Somewhere darkness falls.

Regards, WM

zelos...@gmail.com schrieb am Freitag, 25. Februar 2022 um 06:31:03 UTC+1:
> torsdag 24 februari 2022 kl. 13:59:32 UTC+1 skrev WM:

> > 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.
> >
> The one you want as "final" is not a matrix.

There is no final matrix visible. You can find every lit fraction in the first column. You will never find a lit fraction residing permanently outside of the first column. But you will never find an empty place in the matrix.

You can avoid the problem by gathering the fractions
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
somewhere else, in another place, outside of the matrix.
Then you get the impression that nothing will remain in the matrix. Apparently.
If you don't look too close, Cantor seems right. But is he right then? Are the remaining fractions, like electrons in quantum theory, not there if not being observed? That is not what a mathematician should believe.

Regards, WM

Chris M. Thomasson pretended :
> On 2/24/2022 4:54 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.
>
> 1, 2, 3, ... can be used to index every fraction.

I disagree. We should be talking about rational numbers, not fractions.

On Friday, February 25, 2022 at 12:49:55 PM UTC+1, FromTheRafters wrote:
> Chris M. Thomasson pretended :
> >
> > 1, 2, 3, ... can be used to index every fraction.
> >
> I disagree.

Really? (Actually, I don't think so.)

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

> We should be talking about rational numbers, not fractions.

Why?

On 2/25/2022 5:34 AM, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 25. Februar 2022 um 06:31:03 UTC+1:
>> torsdag 24 februari 2022 kl. 13:59:32 UTC+1 skrev WM:
>
>>> 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.
>>>
>> The one you want as "final" is not a matrix.
>
> There is no final matrix visible. You can find every lit fraction in the first column. You will never find a lit fraction residing permanently outside of the first column. But you will never find an empty place in the matrix.
>
> You can avoid the problem by gathering the fractions
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> somewhere else, in another place, outside of the matrix.
> Then you get the impression that nothing will remain in the matrix. Apparently.
> If you don't look too close, Cantor seems right. But is he right then? Are the remaining fractions, like electrons in quantum theory, not there if not being observed? That is not what a mathematician should believe.
>
> Regards, WM

that is crazy talk.

you are not a mathematician.

On 2/25/2022 5:24 AM, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 24. Februar 2022 um 23:25:34 UTC+1:
>> I am not interested in the sequence of fractions; I am interested in the individual elements of the matrices M(k).
>>

> Somewhere darkness falls.
>
> Regards, WM

Your darkness is your lack of knowing algebra.

Fritz Feldhase laid this down on his screen :
> On Friday, February 25, 2022 at 12:49:55 PM UTC+1, FromTheRafters wrote:
>> Chris M. Thomasson pretended :
>>>
>>> 1, 2, 3, ... can be used to index every fraction.
>>>
>> I disagree.
>
> Really? (Actually, I don't think so.)
>
> See: https://en.wikipedia.org/wiki/Pairing_function
>
>> We should be talking about rational numbers, not fractions.
>
> Why?

Because the topic is about enumerating the positive rationals. Reals
have an integer part and a 'fractional' part but are not necessarily in
Q.

On Friday, February 25, 2022 at 3:40:12 PM UTC+1, FromTheRafters wrote:

> > > We should be talking about rational numbers, not fractions.
> > >
> > Why?
> >
> Because the topic is about enumerating the positive rationals. [...]

Huh?! Actually WM is discussing the countability of the positive fractions here. Didn't you notice that?

<quote>
> Exchanging places leaves all fractions within the matrix, [...]
> all [...] 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."
</quote>

Moreover Cantor's pairing fuction (quoted by WM) does not map Q onto IN but IN x IN.

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

So yes: "[The numbers] 1, 2, 3, ... can be used to index every fraction."

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

On Friday, February 25, 2022 at 3:57:43 PM UTC+1, Fritz Feldhase wrote:
> On Friday, February 25, 2022 at 3:40:12 PM UTC+1, FromTheRafters wrote:
> > > >
> > > > We should be talking about rational numbers, not fractions.
> > > >
> > > Why?
> > >
> > Because the topic is about enumerating the positive rationals. [...]
> >
> Huh?! Actually WM is discussing the countability of the positive fractions here. Didn't you notice that?
>
> <quote>
> > Exchanging places leaves all fractions within the matrix, [...]
> > all [...] 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."
> </quote>

Or:

<quote>
> 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.
</quote>

So no, the topic is NOT about enumerating the positive rationals, but about enumerating the positive fractions.

On 2/25/2022 8:57 AM, Fritz Feldhase wrote:
> On Friday, February 25, 2022 at 3:40:12 PM UTC+1, FromTheRafters wrote:
>
>>>> We should be talking about rational numbers, not fractions.
>>>>
>>> Why?
>>>
>> Because the topic is about enumerating the positive rationals. [...]
>
> Huh?! Actually WM is discussing the countability of the positive fractions here. Didn't you notice that?
>
> <quote>
>> Exchanging places leaves all fractions within the matrix, [...]
>> all [...] 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."
> </quote>
>
> Moreover Cantor's pairing fuction (quoted by WM) does not map Q onto IN but IN x IN.
>
> See: https://en.wikipedia.org/wiki/Pairing_function
>
> So yes: "[The numbers] 1, 2, 3, ... can be used to index every fraction."
>
> See: https://en.wikipedia.org/wiki/Pairing_function#Inverting_the_Cantor_pairing_function

Rationals

p/q where q =/= 0, and p and q are integers

Fractions
p/q where q =/= 0 and p and q are whole numbers

both rationals and fractions are real numbers

every fraction is a rational,

you guys are only arguing about the - sign

is WMs insanity spreading ?

On 2/25/2022 9:02 AM, Fritz Feldhase wrote:
> On Friday, February 25, 2022 at 3:57:43 PM UTC+1, Fritz Feldhase wrote:
>> On Friday, February 25, 2022 at 3:40:12 PM UTC+1, FromTheRafters wrote:
>>>>>
>>>>> We should be talking about rational numbers, not fractions.
>>>>>
>>>> Why?
>>>>
>>> Because the topic is about enumerating the positive rationals. [...]
>>>
>> Huh?! Actually WM is discussing the countability of the positive fractions here. Didn't you notice that?
>>
>> <quote>
>>> Exchanging places leaves all fractions within the matrix, [...]
>>> all [...] 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."
>> </quote>
>
> Or:
>
> <quote>
>> 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.
> </quote>
>
> So no, the topic is NOT about enumerating the positive rationals, but about enumerating the positive fractions.

LAMO ! which positive fraction is not a positive rational ?

On Friday, February 25, 2022 at 4:09:34 PM UTC+1, sergio wrote: [...]

It's slightly more complecated.

Rationals: 1/1 = 2/2.

/Fractions/: 1/1 =/= 2/2.

Of course using the same notation for /fractions/ and rationals may be misleading, I admit.

Hence writing (n, m) for /fractions/ might be a good idea.

Then it's not really a surprise that (1, 1) =/= (2, 2), right?

It seems to me that here terminology may differ. Some will consider /set of fractions/ c /set of rationals/. I guess. That's NOT the case for the /fractions/ WM is referring to.

On Friday, February 25, 2022 at 4:15:54 PM UTC+1, sergio wrote:

> LAMO ! which positive fraction is not a positive rational?

You are an idiot.

From the After wrote: "We should be talking about rational numbers, not fractions."

But actually, WM is talking about /fractions/ n/m, where, say, 1/1 =/= 2/2.

That's clearly not the case if we are denoting rational numbers with "n/m". Here we have 1/1 = 2/2.

On 2/25/2022 9:29 AM, Fritz Feldhase wrote:
> On Friday, February 25, 2022 at 4:15:54 PM UTC+1, sergio wrote:
>
>> LAMO ! which positive fraction is not a positive rational?
>
> You are an idiot.
>
> From the After wrote: "We should be talking about rational numbers, not fractions."
>
> But actually, WM is talking about /fractions/ n/m, where, say, 1/1 =/= 2/2.
>
> That's clearly not the case if we are denoting rational numbers with "n/m". Here we have 1/1 = 2/2.

that is still bogus.

You are saying WM "does not include all fractions not in proper form"

1/10 = 10/100 = 100/1000 so the latter two are not included by WM? , although they are included by Cantor, and also by WM in his matrix of illusions

On 2/25/2022 9:25 AM, Fritz Feldhase wrote:
> On Friday, February 25, 2022 at 4:09:34 PM UTC+1, sergio wrote: [...]
>
> It's slightly more complecated.
>
> Rationals: 1/1 = 2/2.
>
> /Fractions/: 1/1 =/= 2/2.
>
> Of course using the same notation for /fractions/ and rationals may be misleading, I admit.
>
> Hence writing (n, m) for /fractions/ might be a good idea.
>
> Then it's not really a surprise that (1, 1) =/= (2, 2), right?
>
> It seems to me that here terminology may differ. Some will consider /set of fractions/ c /set of rationals/. I guess. That's NOT the case for the /fractions/ WM is referring to.

that is if one invokes proper form, but it is not in enumeration.

On Friday, February 25, 2022 at 4:56:41 PM UTC+1, sergio wrote:

> You are saying WM "does not include all fractions not in proper form"

No, I didn't say that.

> 1/10 = 10/100 = 100/1000 so the latter two are not included by WM, although they are included by Cantor, and also by WM in his matrix of illusions

No, 1/10, 10/100, 100/1000 are elements in Cantor's sequence and WM's matrix. But (since we are laking about /fractions/) they are pairwise distinct.

This allows for a bijection from {n, m, o} c IN onto {1/10, 10/100, 100/1000} where n, m, o are pairwise distinct (i. e. card({n, m, o}) = 3).

On Friday, February 25, 2022 at 4:58:23 PM UTC+1, sergio wrote:
> On 2/25/2022 9:25 AM, Fritz Feldhase wrote:
> > On Friday, February 25, 2022 at 4:09:34 PM UTC+1, sergio wrote: [...]
> >
> > It's slightly more complecated.
> >
> > Rationals: 1/1 = 2/2.
> >
> > /Fractions/: 1/1 =/= 2/2.
> >
> > Of course using the same notation for /fractions/ and rationals may be misleading, I admit.
> >
> > Hence writing (n, m) for /fractions/ might be a good idea.
> >
> > Then it's not really a surprise that (1, 1) =/= (2, 2), right?
> >
> that is if one invokes proper form [...]

[ For the sake of the argument let's agree to write "n/m" instead of "(n, m)", when talking about /fractions/. ]

Actually, we may _define_ the pos. rational numbers as equivalence classes of /fractions/.

The equivalence relation adopted here is defined by a/b ~ c/d <-> a*d = c*b.

Then [1/1] := {1/1, 2/2, 3/3, ...}, [1/2] := {1/2, 2/4, 3/6, ...}, etc.

and we have/get, say, [1/1] = [2/2], etc. with [a/b] := the equivalence class with fraction a/b as a member

Of course, we usually prefer to write (just) a/b instead of [a/b]. :-P

Fritz Feldhase schrieb am Freitag, 25. Februar 2022 um 17:07:36 UTC+1:
> On Friday, February 25, 2022 at 4:56:41 PM UTC+1, sergio wrote:
>
> > You are saying WM "does not include all fractions not in proper form"
> No, I didn't say that.
>
> > 1/10 = 10/100 = 100/1000 so the latter two are not included by WM, although they are included by Cantor, and also by WM in his matrix of illusions
>
> No, 1/10, 10/100, 100/1000 are elements in Cantor's sequence and WM's matrix. But (since we are laking about /fractions/) they are pairwise distinct.

Thanks for your explanations. You have surpassed them all. Why not do the last step?

Regards, WM

horand....@gmail.com schrieb am Freitag, 25. Februar 2022 um 17:25:27 UTC+1:

> The sequences {m_{i,j}(k):k \in IN} are sequences of rational numbers with well-defined formation rules. (The definitions are recursive, but they are finite.) Does the sequence {1/2, 2/1, 3/1, 3/1, 6/1, ...} have a limit or not?

The sequence does not have a limit like 1, 2, 3, ... does not have a limit. Every place of the matrix (except 1/1) is occupied by a fraction that before had occupied another place. If the matrix

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, ...
....

contains only fractions before shuffling them, then all shuffled fractions are fractions too after all shuffling. Note that ω and ω/1 and ω/2 and ... and 1/ω and 2/ω and ... are not fractions and therefore are not in the matrix of fractions which are to be enumerated. Not before and not after. Enumerating has nothing, really, really nothing to do with limits.

Regards, WM