måndag 19 september 2022 kl. 18:57:50 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 19. September 2022 um 09:22:12 UTC+2:
> > fredag 16 september 2022 kl. 09:38:33 UTC+2 skrev WM:
>
> > > Axiom No.1: Inclusion-monotonic sequences of sets cannot have an empty intersection unless they have an empty term.
> > This one is false and it is trivial to prove.
> In a foolish theory. But I do not accept foolish theories.

Nope, in REAL mathematics. You do not accept REAL mathematics.

> > > Axiom No. 2: Exchanging an X and an O never deletes one of them.
> > > More is not required to show matheology wrong.
> > It is trivial to show you are wrong though.
> In a foolish theory. But I do not accept foolish theories.

You accept foolish theories, your stuff IS THE FOOLISH SHIT!

>
> Regards, WM

FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> WM expressed precisely :
> > FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> >> It happens that WM formulated :
> >
> >>> I understand: Every fraction can be indexed. That amounts to removing all
> >>> O's.
> >> No, it doesn't.
> >
> > It is so by definition.
> No it isn't.

Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
Conclusion: All fractions are indexed <==> All O's are removed.

Regards, WM

On 9/20/2022 8:42 AM, WM wrote:
> FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
>> WM expressed precisely :
>>> FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
>>>> It happens that WM formulated :
>>>
>>>>> I understand: Every fraction can be indexed. That amounts to removing all
>>>>> O's.
>>>> No, it doesn't.
>>>
>>> It is so by definition.
>> No it isn't.
>
> Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
> Conclusion: All fractions are indexed <==> All O's are removed.

your Definition is wrong, it is an Assumption

and your conclusion is wrong, all O's are replaced by an X.

>
> Regards, WM

if you wonder about where the O's went, then wonder about where all the X's come from.

WM and his Silly Math games.

WM was thinking very hard :
> FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
>> WM expressed precisely :
>>> FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
>>>> It happens that WM formulated :
>>>>> I understand: Every fraction can be indexed. That amounts to removing all
>>>>> O's.
>>>> No, it doesn't.
>>>
>>> It is so by definition.
>> No it isn't.
>
> Definition: An indexed fraction is denoted by X.

Then there is an X in each cell.

> A not indexed fraction is denoted by O.

Sorry, there's no room for the Os -- no double-occupancy

> Conclusion: All fractions are indexed <==> All O's are removed.

It simply started out all Xs and remained so.

On Monday, 19 September 2022 at 14:05:43 UTC-3, WM wrote:
[...]
> Everybody can see that you are wrong.

As long as you think you are everybody, this may well be true. Fortunately almost every body (your poor students excepted) can see that I am right on this point, and you are the idiot.

WM <askasker48@gmail.com> writes:

> Ben Bacarisse schrieb am Freitag, 16. September 2022 um 18:57:35 UTC+2:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>> > Ben Bacarisse schrieb am Donnerstag, 15. September 2022 um 01:47:38 UTC+2:
>> >> WM <askas...@gmail.com> writes:
>> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> >> Unendlichen" at Technische Hochschule Augsburg.)
>> >
>> >> >> But maybe one of his students will catch a post or two.
>> >>
>> >> How do propose to present my argument since you could not understand it?
>> >
>> > Chuckle. I could present your arguments as follows:
>> >
>> > The topic is that never an O is removed.
>> > BB: That's a silly topic because no one disputes it.
>> > BB: Your OP is full of waffle. I had to turn it into maths to explain how
>> > the Os "disappear".
>> >
>> > But, no I will not present that.
>> Why not? It's all true.
>
> What should I say? You did not explain how the O's will disappear.

Ah, so you didn't understand the argument I made. You said you did.

--
Ben.

tisdag 20 september 2022 kl. 15:42:56 UTC+2 skrev WM:
> FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> > WM expressed precisely :
> > > FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> > >> It happens that WM formulated :
> > >
> > >>> I understand: Every fraction can be indexed. That amounts to removing all
> > >>> O's.
> > >> No, it doesn't.
> > >
> > > It is so by definition.
> > No it isn't.
> Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
> Conclusion: All fractions are indexed <==> All O's are removed.
>
> Regards, WM

False

On Wednesday, September 21, 2022 at 3:30:23 AM UTC+2, Ben Bacarisse wrote:
> WM <askas...@gmail.com> writes:
>
> > Ben Bacarisse schrieb am Freitag, 16. September 2022 um 18:57:35 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> >> Unendlichen" at Hochschule Augsburg.)
> >> > Ben Bacarisse schrieb am Donnerstag, 15. September 2022 um 01:47:38 UTC+2:
> >> >> WM <askas...@gmail.com> writes:
> >> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> >> >> Unendlichen" at Technische Hochschule Augsburg.)
> >> >
> >> >> >> But maybe one of his students will catch a post or two.
> >> >>
> >> >> How do propose to present my argument since you could not understand it?
> >> >
> >> > Chuckle. I could present your arguments as follows:
> >> >
> >> > The topic is that never an O is removed.
> >> > BB: That's a silly topic because no one disputes it.
> >> > BB: Your OP is full of waffle. I had to turn it into maths to explain how
> >> > the Os "disappear".
> >> >
> >> > But, no I will not present that.
> >> Why not? It's all true.
> >
> > What should I say? You did not explain how the O's will disappear.
> Ah, so you didn't understand the argument I made. You said you did.
>
> --
> Ben.

Is that a surprise?

Mücke has never - and I mean not once - understood an argument that
demonstrated that his blatant nonsense is blatant nonsense. And that
includes the divergent integrals in his bestseller.
In effect, he's been trying to understand the definition of the
natural numbers for 20 years. G-d only knows why. Most people get
along fine without ever hearing of Cantor and Peano et al.

Ben Bacarisse schrieb am Mittwoch, 21. September 2022 um 03:30:23 UTC+2:
> WM <askas...@gmail.com> writes:
>

> > What should I say? You did not explain how the O's will disappear.
> Ah, so you didn't understand the argument I made. You said you did.

Please repeat it so that a student could understand without my intervention: Where do the O's, indicating not indexed fractions, remain? How do they leave the matrix?

Regards, WM

zelos...@gmail.com schrieb am Mittwoch, 21. September 2022 um 08:25:40 UTC+2:
> tisdag 20 september 2022 kl. 15:42:56 UTC+2 skrev WM:
> > FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> > > WM expressed precisely :
> > > > FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> > > >> It happens that WM formulated :
> > > >
> > > >>> I understand: Every fraction can be indexed. That amounts to removing all
> > > >>> O's.
> > > >> No, it doesn't.
> > > >
> > > > It is so by definition.
> > > No it isn't.
> > Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
> > Conclusion: All fractions are indexed <==> All O's are removed.
> >
> False

What do you mean? Not all O's are removed but all fractions are indexed? Or not all fractions are indexed?

Regards, WM

FromTheRafters schrieb am Dienstag, 20. September 2022 um 20:07:39 UTC+2:
> WM was thinking very hard :
> > FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> >> WM expressed precisely :
> >>> FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> >>>> It happens that WM formulated :
> >>>>> I understand: Every fraction can be indexed. That amounts to removing all
> >>>>> O's.
> >>>> No, it doesn't.
> >>>
> >>> It is so by definition.
> >> No it isn't.
> >
> > Definition: An indexed fraction is denoted by X.
> Then there is an X in each cell.
> > A not indexed fraction is denoted by O.
> Sorry, there's no room for the Os -- no double-occupancy
> > Conclusion: All fractions are indexed <==> All O's are removed.
> It simply started out all Xs and remained so.

No, it started out with only all unit fractions indeXed:

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, ...
....

Integer fractions are denoted by X's:

XOOO...
XOOO...
XOOO...
XOOO...
....

In the end, when all exchanges of X and O have been carried through according to Cantor we have

XXXX...
XXXX...
XXXX...
XXXX...
....

but no O has left the matrix. Where are they?

Regards, WM

Regards, WM

On 9/21/2022 9:05 AM, WM wrote:
> Ben Bacarisse schrieb am Mittwoch, 21. September 2022 um 03:30:23 UTC+2:
>> WM <askas...@gmail.com> writes:
>>
>
>>> What should I say? You did not explain how the O's will disappear.
>> Ah, so you didn't understand the argument I made. You said you did.
>
> Please repeat it so that a student could understand without my intervention: Where do the O's, indicating not indexed fractions, remain? How do they leave the matrix?

they were peeled off by your sticky fingers, then you put an X pastie on.

>
> Regards, WM

On 9/21/2022 9:10 AM, WM wrote:
> FromTheRafters schrieb am Dienstag, 20. September 2022 um 20:07:39 UTC+2:
>> WM was thinking very hard :
>>> FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
>>>> WM expressed precisely :
>>>>> FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
>>>>>> It happens that WM formulated :
>>>>>>> I understand: Every fraction can be indexed. That amounts to removing all
>>>>>>> O's.
>>>>>> No, it doesn't.
>>>>>
>>>>> It is so by definition.
>>>> No it isn't.
>>>
>>> Definition: An indexed fraction is denoted by X.
>> Then there is an X in each cell.
>>> A not indexed fraction is denoted by O.
>> Sorry, there's no room for the Os -- no double-occupancy
>>> Conclusion: All fractions are indexed <==> All O's are removed.
>> It simply started out all Xs and remained so.
>
> No, it started out with only all unit fractions indeXed:
>
> 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, ...
> ...
>
> Integer fractions are denoted by X's:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> In the end, when all exchanges of X and O have been carried

Wrong! *you overwrite both Os and Xs* with Xs, and you say so!

and Cantor has nothing to do with it, liar.

> Regards, WM
>
>
> Regards, WM

On Thursday, September 1, 2022 at 7:24:05 AM UTC-7, Ben Bacarisse wrote:
> WM <askas...@gmail.com> writes:
>
> > Ben Bacarisse schrieb am Mittwoch, 31. August 2022 um 17:09:18 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> >> Unendlichen" at Hochschule Augsburg.)
> >> > (1) Cantor has proved that all positive fractions m/n can be
> >> > enumerated by all natural numbers k:
> >> >
> >> > k = (m + n - 1)(m + n - 2)/2 + m. (*)
> >> I.e. k(n,m) is a bijection from NxN to N, a fact provable by any student
> >> who has read your textbook. Do you agree that such a student could
> >> prove this fact?
> >
> > Of course.
> >
> >> > This is tantamount to enumerating the positive fractions by the
> >> > integer fractions of the first column of 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, ...
> >> > ...
> >> >
> >> > Of course also the integer fractions belong to the fractions to be
> >> > enumerated. Therefore his approach is tantamount to exchanging X's and
> >> > O's in the matrix until all O's have disappeared:
> >> >
> >> > X, O, O, O, ...
> >> > X, O, O, O, ...
> >> > X, O, O, O, ...
> >> > X, O, O, O, ...
> >> > ...
> >> It's equivalent to drawing arrows between the Xs and every cell
> >> (including the Xs) so that every cell is at the beginning and end of
> >> just one arrow. You can see how this can be done, yes?
> >
> > Of course. But why do you stop here?
> Because in the past you have denied that there is a bijection between N
> and NxN. This is what NxN (or Q for that matter) being enumerated means
> and I am glad that you agree. I stopped there because I am not
> interested in the non-mathematical parts where things get "removed" from
> matrices. That's a word game, and no fun for me.
>
> At least we agree that NxN and Q are denumerable sets. That's a big
> step for you.
>
> --
> Ben.

Ah, but what structural resources do you need to maintain any particular relation.

Ah, but what structural resource defines all the functions between these smaller and larger sets.

It's the stupidest fallacy of set theory that two quantities being countably infinite
have examples besides existence proofs, of being 1-1 and onto, among the spaces
that are infinite products of spaces, of some countable space.

This is that the rationals are "huge" according to usual constructive bounds,
"huge" as a set, though countable, and that NxN or N^2 "is on the order of
the square", and much "larger" in terms of how big the space of functions is
what particular best arithmetizations project down into the inner.

Then, just like WM is a dunce about related rates, the usual absent
sufficiency of the transfinite cardinality, is a great buffoon.

For a topography of spaces and for topology and surfaces,
the old dunce cap is a cone - set theory these days besides
just all the postmodern and descriptive set theory, ..., also
includes a bunch of elementary set theory in finite combinatorics,
that results what is the "infinite space in sets" of its terms, but
actually "infinite superspaces", about how to talk about sets
and their sizes and what properties of sets result naturally
definitions for "larger" what according to set theory are only
"sets", about the validities where between the models of sets
that also have simple models, or "unique" ones in the entire theory
like "the integers", and where their simplest models in the establishment
of form, make their own validities and relate why otherwise set-theoretic
operations, would use "all the resources of the mathematical universe"
and have none left.

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Please repeat it so that a student could understand without my
> intervention: Where do the O's, indicating not indexed fractions,
> remain? How do they leave the matrix?

1. These matrices are functions from NxN to {0,1} (using 0 for an O and
1 for an X).

2. The initial matrix, M_0, is the function M_0(n,m) = [n = 1]. (Here
[P(x)] is the Iverson bracket: 1 if P(x) and 0 otherwise).

3. A sequence of swaps is defined (at least you started off by talking
about swaps) that produce M_(n+1) from M_n.

4. We swap the 0 at (i,j) for which k(i,j) is minimal with the X at
(p,q) for which k(p,q) is minimal but with k(p,q) > k(i,j).

5. We now need to define what it means when you talk (as you did) about
the result of an endless sequence of swaps. The only meaning of that
phrase is a limit, and in fact the sequence of matrices -- the
function sequence M_n -- is indeed convergent.

6. lim_{n->oo} M_n = M_l where M_l is the function M_l(n,m) = 1. In the
limit, no position has an O.

Notes.

1. If you don't like the definition in 5, you have to tell your students
what you meant by "the result" of an endless sequence of swaps.

2. The matrices can be represented as function of one variable which
makes the swaps easier to visualise and explain, but you did not like
(or didn't follow) that transformation so I have omitted it.

3. Different patters of swap produce different limits. You can swap in
such a way as to have 1, 42 or an infinite number of Os in the limit.

Do you think you could explain this to the class? Is there some part
that you don't understand well enough to explain? If you feel brave
enough, you could explain that all of this works with potentially
infinite sets, and that they can find the explanation of how to
calculate the limit in your textbook.

--
Ben.

"Ross A. Finlayson" <ross.finlayson@gmail.com> writes:

> On Thursday, September 1, 2022 at 7:24:05 AM UTC-7, Ben Bacarisse wrote:

>> Because in the past you have denied that there is a bijection between N
>> and NxN. This is what NxN (or Q for that matter) being enumerated means
>> and I am glad that you agree. I stopped there because I am not
>> interested in the non-mathematical parts where things get "removed" from
>> matrices. That's a word game, and no fun for me.
>>
>> At least we agree that NxN and Q are denumerable sets. That's a big
>> step for you.
>
> Ah, but what structural resources do you need to maintain any
> particular relation.
>
> Ah, but what structural resource defines all the functions between
> these smaller and larger sets.
>
> It's the stupidest fallacy of set theory that two quantities being
> countably infinite have examples besides existence proofs, of being
> 1-1 and onto, among the spaces that are infinite products of spaces,
> of some countable space.

Why did you wait and reply to me rather than replying to WM when he
originally agreed with me? Indeed why did you not reply to WM when he
first posted the formula for the bijection many, many posts ago? Is
there a sort of crank code whereby you all agree not to disagree with
each other?

--
Ben.

On Wednesday, September 21, 2022 at 9:21:20 AM UTC-7, Ben Bacarisse wrote:
> "Ross A. Finlayson" <ross.fi...@gmail.com> writes:
>
> > On Thursday, September 1, 2022 at 7:24:05 AM UTC-7, Ben Bacarisse wrote:
>
> >> Because in the past you have denied that there is a bijection between N
> >> and NxN. This is what NxN (or Q for that matter) being enumerated means
> >> and I am glad that you agree. I stopped there because I am not
> >> interested in the non-mathematical parts where things get "removed" from
> >> matrices. That's a word game, and no fun for me.
> >>
> >> At least we agree that NxN and Q are denumerable sets. That's a big
> >> step for you.
> >
> > Ah, but what structural resources do you need to maintain any
> > particular relation.
> >
> > Ah, but what structural resource defines all the functions between
> > these smaller and larger sets.
> >
> > It's the stupidest fallacy of set theory that two quantities being
> > countably infinite have examples besides existence proofs, of being
> > 1-1 and onto, among the spaces that are infinite products of spaces,
> > of some countable space.
> Why did you wait and reply to me rather than replying to WM when he
> originally agreed with me? Indeed why did you not reply to WM when he
> first posted the formula for the bijection many, many posts ago? Is
> there a sort of crank code whereby you all agree not to disagree with
> each other?
>
> --
> Ben.

You mean hypocrisy?

Because soundness and validity and so on is what not-cranks agree to believe.

All I want is a perfect theory of mathematics resolving all paradox, ....

I'm interested in all aspects foundations, here to tell you that
it's quite amazing the completeness theorems of mathematics.

Then as should be clear to any who have any critical understanding at all,
my approach isn't retro-finitism, though, it is retro-classical, retro super-classical.

Then, WM is all "explained" and dispatched and set theory's "paradoxes"
are "explained" and dispatched, these weaknesses of the naive arguments
to other naive arguments end "explained" and dispatched, really my goal
is to have sat through and more or less commented an education for the lot of you.

So I don't need one, ..., "something better".

Ben Bacarisse schrieb am Mittwoch, 21. September 2022 um 18:10:59 UTC+2:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > Please repeat it so that a student could understand without my
> > intervention: Where do the O's, indicating not indexed fractions,
> > remain? How do they leave the matrix?

> 6. lim_{n->oo} M_n = M_l where M_l is the function M_l(n,m) = 1. In the
> limit, no position has an O.

That is not an explanation, but a counterfactual and rather silly claim. (By the way limits are no suitable to prove indexing which is a stepwise process.) As long as there is no way for an O to be removed from the matrix, your claim cannot be accepted. (Hopefully I will never meet a student lacking logical thinking to the extent required to accept your claim.)

> 1. If you don't like the definition in 5, you have to tell your students
> what you meant by "the result" of an endless sequence of swaps.

The result is what Cantor claimed it to be and what is needed if the enumeration shall be completed: All indices (all X) have settled at their final positions.
>
> 2. The matrices can be represented as function of one variable which
> makes the swaps easier to visualise and explain, but you did not like
> (or didn't follow) that transformation so I have omitted it.

All you have to explain is how an O leaves the matrix.
>
> 3. Different patters of swap produce different limits. You can swap in
> such a way as to have 1, 42 or an infinite number of Os in the limit.

Try to understand: As long as swaps are concerned, no O will leave the matrix or any linear structure.
>
> Do you think you could explain this to the class? Is there some part
> that you don't understand well enough to explain?

I understand that you are claiming what you cannot support by logic: As long as swaps are concerned, no O will leave.

> you could explain that all of this works with potentially
> infinite sets,

Of course. But that is not the topic. You have not understood but you boast to have the solution. You have not.

Regards, WM

On 9/21/2022 10:10 AM, WM wrote:

> Integer fractions are denoted by X's:
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> In the end,

There is no end.
There is no exchange after which
all exchanges have been carried through.

> when all exchanges of X and O have been
> carried through

That's different.
There is no last exchange before that point.
If you insert "definable", _no_
there is _no_ last exchange before that point,
definable or (hypothetical) undefinable.

> [...] we have
> XXXX...
> XXXX...
> XXXX...
> XXXX...
> ...
>
> but no O has left the matrix.

_You can't get there from here_

That's a different claim from
"There is no there".

There is 𝑴ₜₕₑᵣₑ
XXXX...
XXXX...
XXXX...
XXXX...
....

( At ⟨row column⟩ = ⟨p q⟩
( there is k/1 where
( k = (p+q-1)(p+q-2)/2+p
( X at q = 1

You can't get to 𝑴ₜₕₑᵣₑ from 𝑴ₕₑᵣₑ
XOOO...
XOOO...
XOOO...
XOOO...
....

( At ⟨row column⟩ = ⟨p q⟩
( there is p/q
( X at q = 1

.... not by any sequence of 2-cell swaps.

| Assume you can get there from here.
| Assume that
| a matrix-sequence ⟨𝑴ₕₑᵣₑ,⋯,𝑴ₜₕₑᵣₑ⟩ exists
| in which, for each split of ⟨𝑴ₕₑᵣₑ,⋯,𝑴ₜₕₑᵣₑ⟩
| there is a last-before matrix 𝑴₋
| there is a first-after matrix 𝑴₊
| and 𝑴₊ = τ(𝑴₋) for some 2-cell swap τ
| | Define 𝓜₋ 𝓜₊ such that
| | 𝑴 is in 𝓜₋ iff
| all 𝑴' ≼ 𝑴 are 𝑴ₕₑᵣₑ-like
| (all fractions occur in 𝑴')
| | 𝑴 is in 𝓜₊ iff
| some 𝑴' ≼ 𝑴 is 𝑴ₜₕₑᵣₑ-like
| (not all fractions occur in 𝑴')
| | 𝓜₋ 𝓜₊ is a split of ⟨𝑴ₕₑᵣₑ,⋯,𝑴ₜₕₑᵣₑ⟩
| each in 𝓜₋ ≺ each in 𝓜₊
| | By assumption,
| 𝑴₋ 𝑴₊ exist,
| 𝑴₋ is last in 𝓜₋
| 𝑴₊ is first in 𝓜₊
| and 𝑴₊ = τ(𝑴₋) for some 2-cell swap τ
| | However,
| 𝑴₋ is in 𝓜₋
| 𝑴₋ must be 𝑴ₕₑᵣₑ-like
| 𝑴₊ is first in 𝓜₊
| all 𝑴 ≺ 𝑴₊ are in 𝓜₋ and 𝑴ₕₑᵣₑ-like
| 𝑴₊ must be 𝑴ₜₕₑᵣₑ-like
| | No 2-cell swap τ exists such that
| 𝑴₋ is 𝑴ₕₑᵣₑ-like
| (all fractions occur in 𝑴₋)
| 𝑴₊ is 𝑴ₜₕₑᵣₑ-like
| (not all fractions occur in 𝑴₊)
| and 𝑴₊ = τ(𝑴₋)
| Contradiction.

Therefore,
you can't get to 𝑴ₜₕₑᵣₑ from 𝑴ₕₑᵣₑ
by any sequence of 2-cell swaps.

That's a different claim from
"There is no 𝑴ₜₕₑᵣₑ"

> Where are they?

In any matrix 𝑴 to which
you can get from 𝑴ₕₑᵣₑ
by a sequence of 2-cell swaps,
all the O's will be in 𝑴

However,
𝑴ₜₕₑᵣₑ is NOT a matrix to which
you can get from 𝑴ₕₑᵣₑ

> Where are they?

Where does the flame go when you
blow out a candle?

Why would it be somewhere?

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Mittwoch, 21. September 2022 um 18:10:59 UTC+2:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>> > Please repeat it so that a student could understand without my
>> > intervention: Where do the O's, indicating not indexed fractions,
>> > remain? How do they leave the matrix?
>
>> 6. lim_{n->oo} M_n = M_l where M_l is the function M_l(n,m) = 1. In the
>> limit, no position has an O.
>
> That is not an explanation, but a counterfactual and rather silly
> claim. (By the way limits are no suitable to prove indexing which is a
> stepwise process.)

Then you need to explain what you mean by the result of an endless
sequence of swaps. Don't blame real mathematicians for having to fill
in the gaps of your vague puzzles.

>> 1. If you don't like the definition in 5, you have to tell your students
>> what you meant by "the result" of an endless sequence of swaps.
>
> The result is what Cantor claimed it to be

You need to say what /you/ meant for anyone to be able to confirm that,
and you also need to cite where Cantor claimed anything about matrices
and swaps. Without both of these your claim that "the result" is what
Cantor claimed is a meaningful as my saying the result is what Bozo the
Clown claimed it to be.

And, of course, you have neither.

> All you have to explain is how an O leaves the matrix.

Don't be silly. You think they went to Florida? Nothing "leaves"
anywhere. A swap produces a new matrix from the previous one and if you
don't say what you meant when you talked about "the result" then all you
have is an endless sequence of matrices. But if you do dare say, you
will find that some swaps have "a result" with no Os in any position.

>> Do you think you could explain this to the class? Is there some part
>> that you don't understand well enough to explain?
>
> I understand that you are claiming what you cannot support by logic:
> As long as swaps are concerned, no O will leave.

Yet "the result", suitably defined, has no position with an O. Anyway,
I don't think you are competent to present this argument so please don't
try, at least not without running the text by me first. Thanks.

--
Ben.

onsdag 21 september 2022 kl. 16:08:06 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 21. September 2022 um 08:25:40 UTC+2:
> > tisdag 20 september 2022 kl. 15:42:56 UTC+2 skrev WM:
> > > FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> > > > WM expressed precisely :
> > > > > FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> > > > >> It happens that WM formulated :
> > > > >
> > > > >>> I understand: Every fraction can be indexed. That amounts to removing all
> > > > >>> O's.
> > > > >> No, it doesn't.
> > > > >
> > > > > It is so by definition.
> > > > No it isn't.
> > > Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
> > > Conclusion: All fractions are indexed <==> All O's are removed.
> > >
> > False
>
> What do you mean? Not all O's are removed but all fractions are indexed? Or not all fractions are indexed?
>
> Regards, WM
Nothing is removed or the likes, the function just EXISTS!

Ben Bacarisse schrieb am Donnerstag, 22. September 2022 um 01:35:47 UTC+2:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > Ben Bacarisse schrieb am Mittwoch, 21. September 2022 um 18:10:59 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> >> Unendlichen" at Hochschule Augsburg.)
> >> > Please repeat it so that a student could understand without my
> >> > intervention: Where do the O's, indicating not indexed fractions,
> >> > remain? How do they leave the matrix?
> >
> >> 6. lim_{n->oo} M_n = M_l where M_l is the function M_l(n,m) = 1. In the
> >> limit, no position has an O.
> >
> > That is not an explanation, but a counterfactual and rather silly
> > claim. (By the way limits are no suitable to prove indexing which is a
> > stepwise process.)
> Then you need to explain what you mean by the result of an endless
> sequence of swaps. Don't blame real mathematicians for having to fill
> in the gaps of your vague puzzles.

I adopt Cantor's interpretation. That is not a vague puzzle but well known to every specialist: The result is obtained when all X's have settles at their final positions such that none does further move. Only then we have a complete use of all indizes.

How can you try to correct me if you don't even know these basic notions???

> >> 1. If you don't like the definition in 5, you have to tell your students
> >> what you meant by "the result" of an endless sequence of swaps.
> >
> > The result is what Cantor claimed it to be
> You need to say what /you/ meant for anyone to be able to confirm that,

I am analyzing Cantor's writings. I jave not made up my own theory. Therefore it is relevant only what Cantor said.
> and you also need to cite where Cantor claimed anything about matrices
> and swaps.

If you can't translate Cantor's writings to the model of matrices, then you should change your occupation. 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, ...
....
with serpent-covering of this matrix appears in all texts for beginners.

> Without both of these your claim that "the result" is what
> Cantor claimed is a meaningful as my saying the result is what Bozo the
> Clown claimed it to be.

Try to think instead of cursing. Instead of starting with the serpent I start with the first column. Should I teach you in smaller steps?
>
> > All you have to explain is how an O leaves the matrix.
> Don't be silly. You think they went to Florida? Nothing "leaves"
> anywhere.

But in your idea of limit all have left nevertheless?

> A swap produces a new matrix from the previous one and if you
> don't say what you meant when you talked about "the result" then all you
> have is an endless sequence of matrices.

That would be true in potential infinity. With step n there is step n+1 and later even step n^n^n. Cantor used actual infinity. All indices have been issued and taken a fixed place in the resulting end.

> But if you do dare say, you
> will find that some swaps have "a result" with no Os in any position.

That is impossible and wrong. No swap will delete an O. But you are right, the O's cannot be found in any definable position - in the end.

> > I understand that you are claiming what you cannot support by logic:
> > As long as swaps are concerned, no O will leave.
> Yet "the result", suitably defined, has no position with an O.

Then either O's have left the matrix by swaps or their positions are not detectable because already at the start dark parts of the matrix were present.. IN fact almost all positions are dark. In fact almost all natural numbers are dark:

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

> Anyway,
> I don't think you are competent to present this argument so please don't
> try, at least not without running the text by me first. Thanks.

If you were an expert, you would know at least the infinity used by Cantor and his "final result".

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 22. September 2022 um 08:00:09 UTC+2:
> onsdag 21 september 2022 kl. 16:08:06 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 21. September 2022 um 08:25:40 UTC+2:
> > > tisdag 20 september 2022 kl. 15:42:56 UTC+2 skrev WM:
> > > > FromTheRafters schrieb am Montag, 19. September 2022 um 19:40:58 UTC+2:
> > > > > WM expressed precisely :
> > > > > > FromTheRafters schrieb am Montag, 19. September 2022 um 00:26:58 UTC+2:
> > > > > >> It happens that WM formulated :
> > > > > >
> > > > > >>> I understand: Every fraction can be indexed. That amounts to removing all
> > > > > >>> O's.
> > > > > >> No, it doesn't.
> > > > > >
> > > > > > It is so by definition.
> > > > > No it isn't.
> > > > Definition: An indexed fraction is denoted by X. A not indexed fraction is denoted by O.
> > > > Conclusion: All fractions are indexed <==> All O's are removed.
> > > >
> > > False
> >
> > What do you mean? Not all O's are removed but all fractions are indexed? Or not all fractions are indexed?
> >

> Nothing is removed or the likes, the function just EXISTS!

And the O's are existing too. The result is when all X have settles at fixed positions. But every O that exists initially in the matrix

XOOO...
XOOO...
XOOO...
XOOO...
....

existst there in the end.

XXXX...
XXXX...
XXXX...
XXXX...
....

Regards, WM

Jim Burns schrieb am Mittwoch, 21. September 2022 um 23:40:50 UTC+2:
> On 9/21/2022 10:10 AM, WM wrote:
>
> > Integer fractions are denoted by X's:
> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...
> >
> > In the end,
> There is no end.

According to Cantor there is completion. That means that all X have settled at final destiny and all O have disappeared (from the visible matrix, which Cantor erroneously considered the whole matrix).

> There is no exchange after which
> all exchanges have been carried through.

Right!

> > when all exchanges of X and O have been
> > carried through
> That's different.
> There is no last exchange before that point.
> If you insert "definable", _no_
> there is _no_ last exchange before that point,
> definable or (hypothetical) undefinable.

Of course there is the end. Before that end there must have happened something (otherwise we would not have reached the end). But as you say there is no last step. There are infinitely many "last steps" which cannot be distinguished because they are dark.
>
> > Where are they?
>
> Where does the flame go when you
> blow out a candle?

That is not an exchange.
>
> Why would it be somewhere?

Because exchanging differs from deleting.

Regards, WM

On 9/22/2022 10:23 AM, WM wrote:
>
> Jim Burns schrieb am Mittwoch, 21. September 2022 um 23:40:50 UTC+2:
>> On 9/21/2022 10:10 AM, WM wrote:
>>
>>> Integer fractions are denoted by X's:
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>>>
>>> In the end,
>> There is no end.
>
> According to Cantor there is completion. That means that all X have settled at final destiny and all O have disappeared (from the visible matrix, which Cantor erroneously considered the whole matrix).
>
>
>> There is no exchange after which
>> all exchanges have been carried through.
>
> Right!
>
>>> when all exchanges of X and O have been
>>> carried through
>> That's different.
>> There is no last exchange before that point.
>> If you insert "definable", _no_
>> there is _no_ last exchange before that point,
>> definable or (hypothetical) undefinable.
>
> Of course there is the end. Before that end there must have happened something (otherwise we would not have reached the end). But as you say there is no last step. There are infinitely many "last steps" which cannot be distinguished because they are dark.
>>
>>> Where are they?
>>
>> Where does the flame go when you
>> blow out a candle?
>
> That is not an exchange.
>>
>> Why would it be somewhere?
>
> Because exchanging differs from deleting.

and you are overwriting O's with X's

you are not exchanging them.

>
> Regards, WM