On Friday, 2 September 2022 at 09:23:22 UTC-7, WM wrote:
> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
> > WM <askas...@gmail.com> writes:
> >
> > > That means there is a bijection between visible natnumbers
> > > and visible fractions.
> > Your textbook does not explain what a visible number is
> My textbook, like claisscal maths deals only with visible numbers.

I think you need to cease introducing new names.

Numbers are not visible because a number is not something that can be seen.

Realisable and non-realisable numbers are better terms than "visible" and "dark".

You make a mockery of yourself every time you use such terms and reduce the credibility of your otherwise good arguments.

Oh, and the junk concept of "infinity" cannot be interpreted as "direction" because direction is a well-defined word:

A path or course along which something moves.

Please, we already have enough charlatans and cranks in the mainstream Church of Mathematics academia. Use words that explain what you mean.

> No reason to mention this. Further in 2015 when it was published I did not know about dark numbers.
> > so I don't know
> > what you mean.
> But you know and agree that all visible places of the matrix are occupied by X's and nevertheless no O disappears from the matrix?
>
> Regards, WM

Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:42:47 UTC+2:
> WM <askas...@gmail.com> writes:
>
> > Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >>
> >> > That means there is a bijection between visible natnumbers
> >> > and visible fractions.
> >> Your textbook does not explain what a visible number is
> >
> > My textbook, like claisscal maths deals only with visible numbers. No
> > reason to mention this. Further in 2015 when it was published I did
> > not know about dark numbers.
> I have always been talking about the maths in your book. Did I not make
> that clear enough? Sorry.
>
> If your book is no longer adequate for this task, please say so, but I
> hope it is because you are not good at answering direct questions, but
> the book is there for all to see. What I wanted to know is:
>
> (a) Is k (now cut) a function from NxN to N as the set N and the term
> function are defined in your book?

In my book there are, as I clearly stated in the introduction, only potentially infinite sets. That are the visible numbers which are sufficient to do classical mathematics.

> (b) Is k surjective (as defined in your book)?
>
> (c) Is k injective (as defined in your book)?
>
Of course.

But Cantor talked about actual infinity. I notice that you don't wish to consider Cantor's mapping. There we have the complete matrix of positive fractions and the complete column of integer fractions:

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, ...
....

Cantor's approach is modelled by exchanging X's and O's in

XOO...
XOO...
XOO...
....

until all O's have disappeared. Do you agree? Or did I misinterpret Cantor? Or do you prefer not to get involved in that question because it could be too dangerous to become a heretic matheologian? I could understand your hesitation. Even bold mathematicians like Doron Zeilberger could not resist the strong forces of the imperialists of matheology.

Regards, WM

On 9/2/2022 3:31 PM, WM wrote:
> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:42:47 UTC+2:
>> WM <askas...@gmail.com> writes:
>>
>>> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
>>>> WM <askas...@gmail.com> writes:
>>>>
>>>>> That means there is a bijection between visible natnumbers
>>>>> and visible fractions.
>>>> Your textbook does not explain what a visible number is
>>>
>>> My textbook, like claisscal maths deals only with visible numbers. No
>>> reason to mention this. Further in 2015 when it was published I did
>>> not know about dark numbers.
>> I have always been talking about the maths in your book. Did I not make
>> that clear enough? Sorry.
>>
>> If your book is no longer adequate for this task, please say so, but I
>> hope it is because you are not good at answering direct questions, but
>> the book is there for all to see. What I wanted to know is:
>>
>> (a) Is k (now cut) a function from NxN to N as the set N and the term
>> function are defined in your book?
>
> In my book there are, as I clearly stated in the introduction, only potentially infinite sets. That are the visible numbers which are sufficient to do classical mathematics.

so your book does not apply to "modern" math, but only "visible" Ants.

> Or did I misinterpret Cantor?

yes, it is the one thing you do consistantly

> Regards, WM

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

> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:42:47 UTC+2:
>> WM <askas...@gmail.com> writes:
>>
>> > Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
>> >> WM <askas...@gmail.com> writes:
>> >>
>> >> > That means there is a bijection between visible natnumbers
>> >> > and visible fractions.
>> >> Your textbook does not explain what a visible number is
>> >
>> > My textbook, like claisscal maths deals only with visible numbers. No
>> > reason to mention this. Further in 2015 when it was published I did
>> > not know about dark numbers.
>> I have always been talking about the maths in your book. Did I not make
>> that clear enough? Sorry.
>>
>> If your book is no longer adequate for this task, please say so, but I
>> hope it is because you are not good at answering direct questions, but
>> the book is there for all to see. What I wanted to know is:
>>
>> (a) Is k (now cut) a function from NxN to N as the set N and the term
>> function are defined in your book?
>
> In my book there are, as I clearly stated in the introduction, only
> potentially infinite sets. That are the visible numbers which are
> sufficient to do classical mathematics.
>
>> (b) Is k surjective (as defined in your book)?
>>
>> (c) Is k injective (as defined in your book)?
>>
> Of course.

Great. So k /is/ a bijection (with everything -- functions, bijections,
N, NxN -- as defined as in your book).

> But Cantor talked about actual infinity. I notice that you don't wish
> to consider Cantor's mapping.

Let's see how far we can get just with WMaths. The bijection you wrote

k(n,m) = (m + n - 1)(m + n - 2)/2 + m

will let us write the effect of a sequence of swaps in terms of a
sequence of WMaths functions. We may hit the rocks and find that WMaths
can not take us where we want to go, but let's try. Are you up for
that?

> There we have the complete matrix of positive fractions and the
> complete column of integer fractions:
>
> 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, ...
> ...
>
> Cantor's approach is modelled by exchanging X's and O's in
>
> XOO...
> XOO...
> XOO...
> ...
>
> until all O's have disappeared. Do you agree?

Here are the cells as numbered by your bijective mapping:

1, 3, 6, 10, 15, ...
2, 5, 9, 14, 20, ...
4, 8, 13, 19, 26, ...
7, 12, 18, 25, 33, ...
11, 17, 24, 32, 41, ...
...

With Os at 3, 5, 6, 8, 9 and so on. Let's use 1 and 0 for Xs and Os so
the matrix can be written as a function from N to {0,1} (as defined in
your book). Since these are functions of N, let's just write the first
few values rather that trying to find the ever-so messy formulas for
where there are 1s and 0s:

M_0(n) = 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, ...

Now what are the first few swaps? The logical choice would be to swap
the first 0 with the first following 1 so that

M_1(n) = 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, ...
M_2(n) = 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, ...
M_2(n) = 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...
...

I think this sequence of functions makes the pattern nice ans clear and
all the functions are just the sort of functions you manipulate in your
book. Would you like these turned back into X and O matrices? I don't
think it helps, so I haven't bothered.

Now the only meaning for the endless swapping that makes sense to me is
a limit, and fortunately you tell your students how to prove that

lim_{n->oo} M_n = M_1

where M_1 is the constant function M_1(k) = 1. You give a nice
definition of convergence for sequences of functions in your textbook,
and the function sequence M_n does indeed converge to that limit.

> Or did I misinterpret Cantor?

I don't know. Do you have a citation for his discussion of this matrix?
I does not have the neat structure of most of his arguments so I don't
think you've got it right.

> Or do you prefer not to get involved in that question because it could
> be too dangerous to become a heretic matheologian?

I'm not sure what you think is dangerous. The WMaths limit of the
WMaths function sequence M_n has no 0s in it, and that seems like the
only reasonable meaning for the effect of an endless sequence of swaps.
It seems that WMaths is up to the task. If that makes me a heretic, I
really don't care.

> I could understand your hesitation.

My only hesitancy was due to your flip-flopping about what is and is not
provable. If k (as you defined it) is not a bijection (as you define
the term in your book) between NxN (as you define it in your book) and N
(as defined in your book) then I could not even start formalising, in
WMaths, what this endless sequence of swaps might look like. Even now,
a feel sure a flip or a flop coming on... You find it hard to stick to
talking about WMaths for some reason.

But I note that /you/ still hesitate to show how a function is proved to
be bijective in WMaths. Well, hesitate is a rather weak term because I
have been asking, on and off, for /years/. It's almost as if you don't
know or are, for some reason, afraid to reveal how simple it is in
WMaths. I'm not expecting your hesitation to vanish, but I live in
hope so I am asking again.

--
Ben.

Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:42:47 UTC+2:
> WM <askas...@gmail.com> writes:
>
> > Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >>
> >> > That means there is a bijection between visible natnumbers
> >> > and visible fractions.
> >> Your textbook does not explain what a visible number is
> >
> > My textbook, like claisscal maths deals only with visible numbers. No
> > reason to mention this. Further in 2015 when it was published I did
> > not know about dark numbers.
> I have always been talking about the maths in your book. Did I not make
> that clear enough? Sorry.

I have been talking about Cantor's enumeration of fractions. My book does nt mention that with good reason.
>
> If your book is no longer adequate for this task,

My book, like all reasonable mathematics, is a good basis to discuss Cantor's claim. That is done here. But the basic assumption that Cantor has made must be accepted in order to discuss it.

Regards, WM

Ben Bacarisse schrieb am Samstag, 3. September 2022 um 00:08:10 UTC+2:
> WM <askas...@gmail.com> writes:

> > There we have the complete matrix of positive fractions and the
> > complete column of integer fractions:
> >
> > 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, ...
> > ...
> >
> > Cantor's approach is modelled by exchanging X's and O's in
> >
> > XOO...
> > XOO...
> > XOO...
> > ...
> >
> > until all O's have disappeared. Do you agree?
> Here are the cells as numbered by your bijective mapping:
>
> 1, 3, 6, 10, 15, ...
> 2, 5, 9, 14, 20, ...
> 4, 8, 13, 19, 26, ...
> 7, 12, 18, 25, 33, ...
> 11, 17, 24, 32, 41, ...
> ...

Yes, all visible cells are filled with X's as Cantor has taught us.
>
> With Os at 3, 5, 6, 8, 9 and so on.

No, the above matrix does not contain O's any longer. You have already indexed all visible cells. That means they contain X's. But you have not applied Cantor's formula correctly.

Here is it again: k = (m + n - 1)(m + n - 2)/2 + m yields the sequence of indexed fractions:
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

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 first 3 steps we get the following matrices

XXOO...
OOOO...
XOOO...
XOOO...
....

XXOO...
XOOO...
OOOO...
XOOO...
....

XXXO...
XOOO...
OOOO...
OOOO...
....

> Now the only meaning for the endless swapping that makes sense to me is
> a limit,

Cantor does not use limits. Nobody would have believed him, if he claimed completeness of enumeration "in the limit".

and fortunately you tell your students how to prove that
>
> lim_{n->oo} M_n = M_1

Here we have no limit but a never disappearing set of O's. That is the question: How can the O's disappear without leaving the matrix!
By the way, every X comes to a position where it has infinitely many O's right to it and below it. If there is a limit, then this limit contains infinitely many O's.

> > Or did I misinterpret Cantor?
> I don't know. Do you have a citation for his discussion of this matrix?

No, the citation is written above. The matrix is but another language, expressing precisely what Cantor said.

> I does not have the neat structure of most of his arguments so I don't
> think you've got it right.

Then think over it again. I apply his formula k = (m + n - 1)(m + n - 2)/2 + m with his result in all visible cells: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... But never an O disappears.

Regards, WM

On 9/3/2022 8:12 AM, WM wrote:
> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:42:47 UTC+2:
>> WM <askas...@gmail.com> writes:
>>
>>> Ben Bacarisse schrieb am Freitag, 2. September 2022 um 18:00:32 UTC+2:
>>>> WM <askas...@gmail.com> writes:
>>>>
>>>>> That means there is a bijection between visible natnumbers
>>>>> and visible fractions.
>>>> Your textbook does not explain what a visible number is
>>>
>>> My textbook, like claisscal maths deals only with visible numbers. No
>>> reason to mention this. Further in 2015 when it was published I did
>>> not know about dark numbers.
>> I have always been talking about the maths in your book. Did I not make
>> that clear enough? Sorry.
>
> I have been talking about Cantor's enumeration of fractions. My book does nt mention that with good reason.
>>
>> If your book is no longer adequate for this task,
>
> My book, like all reasonable mathematics, is a good basis to discuss Cantor's claim. That is done here. But the basic assumption that Cantor has made must be accepted in order to discuss it.
>
> Regards, WM

you use Cantors name over and over again, to try to make your math seem important.

However, your math is too riddled with obvious errors and bad math to ever be serious.

you put your PHD Ego on a subject you know nothing about. [I would replace "nothing" with "little" if you knew algebra, but you do not.]

On 9/3/2022 8:46 AM, WM wrote:
> Ben Bacarisse schrieb am Samstag, 3. September 2022 um 00:08:10 UTC+2:
>> WM <askas...@gmail.com> writes:
>
>>> There we have the complete matrix of positive fractions and the
>>> complete column of integer fractions:
>>>
>>> 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, ...
>>> ...
>>>
>>> Cantor's approach is modelled by exchanging X's and O's in
>>>
>>> XOO...
>>> XOO...
>>> XOO...
>>> ...
>>>
>>> until all O's have disappeared. Do you agree?
>> Here are the cells as numbered by your bijective mapping:
>>
>> 1, 3, 6, 10, 15, ...
>> 2, 5, 9, 14, 20, ...
>> 4, 8, 13, 19, 26, ...
>> 7, 12, 18, 25, 33, ...
>> 11, 17, 24, 32, 41, ...
>> ...
>
> Yes, all visible cells are filled with X's as Cantor has taught us.

distraction => the first column is not your indexes in your matrix, you made mistakes

>
>>> Or did I misinterpret Cantor?
>> I don't know. Do you have a citation for his discussion of this matrix?
>
> No, the citation is written above. The matrix is but another language, expressing precisely what Cantor said.

*Liar*
1. you have no citation to a matrix by Cantor.
2. Your matrix does not agree with Cantor, so you made mistakes.

>
>> I does not have the neat structure of most of his arguments so I don't
>> think you've got it right.
>
> Then think over it again. I apply his formula k = (m + n - 1)(m + n - 2)/2 + m with his result in all visible cells: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... But never an O disappears.

Fix your mistakes

>
> Regards, WM
>

WM <askasker48@gmail.com> writes:

> Ben Bacarisse schrieb am Samstag, 3. September 2022 um 00:08:10 UTC+2:
>> WM <askas...@gmail.com> writes:
>
>> > There we have the complete matrix of positive fractions and the
>> > complete column of integer fractions:
>> >
>> > 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, ...
>> > ...
>> >
>> > Cantor's approach is modelled by exchanging X's and O's in
>> >
>> > XOO...
>> > XOO...
>> > XOO...
>> > ...
>> >
>> > until all O's have disappeared. Do you agree?
>> Here are the cells as numbered by your bijective mapping:
>>
>> 1, 3, 6, 10, 15, ...
>> 2, 5, 9, 14, 20, ...
>> 4, 8, 13, 19, 26, ...
>> 7, 12, 18, 25, 33, ...
>> 11, 17, 24, 32, 41, ...
>> ...
>
> Yes, all visible cells are filled with X's as Cantor has taught us.

No. I've just written some numbers down to illustrate what you
bijective mapping k looks like.

>> With Os at 3, 5, 6, 8, 9 and so on.
>
> No, the above matrix does not contain O's any longer.

I think you'd got lost at the first step. I'm using your mapping to
give write the matrix as a function from N to {0,1}. The initial matrix
has both Xs and Os. In fact it's simply this function:

M_0(i) = 1 if m = 1 and 0 otherwise where (n,m) = k'(i)

Since this is the first step, you really need to be able to see how M_0
represents the initial matrix a WMaths function from N to {0,1}. No
completed anything, everything potential.

>> Now the only meaning for the endless swapping that makes sense to me is
>> a limit,
>
> Cantor does not use limits.

I know. Do you just skim posts for keywords rather than reading them?
I said we can get rid of the Os in WMaths without using anything from
Cantor, just the maths in your book. You seemed to have stopped talking
about WMaths at the first sentence above.

>> lim_{n->oo} M_n = M_1
>
> Here we have no limit but a never disappearing set of O's. That is the
> question: How can the O's disappear without leaving the matrix!

If you want to grasp how it's done, you'd have to stick with nothing but
the maths in your book and you'll have to ask intelligent questions when
you don't follow a step. You seemed to get lost at the very start by
thinking I was talking about something other than your WMaths.

You are not good a being a student, but I could set a sequence of
exercises, based on your book, that would take you through the steps to
see how the Os "disappear" without anything from Cantor -- just WMaths.
Would you like me to do that?

>> > Or did I misinterpret Cantor?
>> I don't know. Do you have a citation for his discussion of this matrix?
>
> No, the citation is written above.

What does "no but it's above mean"? You know what a citation is? Just
say if you made it up..

--
Ben.

Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:

> I said we can get rid of the Os in WMaths without using anything from
> Cantor, just the maths in your book.

And that is clearly wrong because, according to every kind of mathematics, never an O leaves the matrix. The only change of the initial matrix

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

consists of exchanging an O and an X. Exchanging two symbols does not delete any of them.

> I could set a sequence of
> exercises, based on your book, that would take you through the steps to
> see how the Os "disappear" without anything from Cantor -- just WMaths.
> Would you like me to do that?

Please try it.

> >> > Or did I misinterpret Cantor?
> >> I don't know. Do you have a citation for his discussion of this matrix?
> >
> > No, the citation is written above.
> What does "no but it's above mean"?

Cantor said that all fractions can be indexed. That means translated to my model that all O's disappear. Can you follow that thought?

Regards, WM

On Sunday, 4 September 2022 at 07:13:35 UTC-3, WM wrote:
> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:
>
> > I said we can get rid of the Os in WMaths without using anything from
> > Cantor, just the maths in your book.
> And that is clearly wrong because, according to every kind of mathematics, never an O leaves the matrix. The only change of the initial matrix
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
> consists of exchanging an O and an X. Exchanging two symbols does not delete any of them.

Nothing but your usual bullshit, in other words. Aside from "Because I (WM) say so", you got nothing. In particular, you have no clue about basic logic. In order to prove an inconsistency in ZFC (or any other axiom system) it is simply not enough to assume the negation of one of the axioms to derive an inconsistency. In your case, assuming that infinity (ω, or any of the alephs) does not exist is not enough to show that ZFC is inconsistent. If you had two firing synapses left, you'd understand that that. So it seems best that you just shut the fuck up and piss off.

On 9/4/2022 5:13 AM, WM wrote:
> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:
>
>> I said we can get rid of the Os in WMaths without using anything from
>> Cantor, just the maths in your book.
>
> And that is clearly wrong because, according to every kind of mathematics, never an O leaves the matrix. The only change of the initial matrix
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> consists of exchanging an O and an X.

>
> Cantor said that all fractions can be indexed. That means translated to my model that all O's disappear. Can you follow that thought?

Wrong, *Cantor Proved all fractions can be indexed using math*.

You cannot un-prove that.

So YOU have made mistakes in your model, and *your model is not based in Math at all*. Fail.

>
> Regards, WM

Gus Gassmann schrieb am Sonntag, 4. September 2022 um 14:39:51 UTC+2:
> On Sunday, 4 September 2022 at 07:13:35 UTC-3, WM wrote:
> > Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:
> >
> > > I said we can get rid of the Os in WMaths without using anything from
> > > Cantor, just the maths in your book.
> > And that is clearly wrong because, according to every kind of mathematics, never an O leaves the matrix. The only change of the initial matrix
> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...
> > consists of exchanging an O and an X. Exchanging two symbols does not delete any of them.
> In particular, you have no clue about basic logic.

I have this understanding of logic: If two symbols are exchanged, then none of them is deleted.

> A In order to prove an inconsistency in ZFC

it issufficient to prove that no O can leave the matrix.

Regards, WM

On Sunday, 4 September 2022 at 10:55:28 UTC-3, WM wrote:
[...]
> I have this understanding of logic: If two symbols are exchanged, then none of them is deleted.

This is correct, but insufficient.

> > A In order to prove an inconsistency in ZFC
>
> it issufficient to prove that no O can leave the matrix.

Ah, no. The fact that the reciprocal 1/n is positive for every natural number n is insufficient to conclude that the limit of the sequence {1, 1/2, 1/3, ..} cannot be 0.

On 9/4/2022 8:55 AM, WM wrote:
> Gus Gassmann schrieb am Sonntag, 4. September 2022 um 14:39:51 UTC+2:
>> On Sunday, 4 September 2022 at 07:13:35 UTC-3, WM wrote:
>>> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:
>>>
>>>> I said we can get rid of the Os in WMaths without using anything from
>>>> Cantor, just the maths in your book.
>>> And that is clearly wrong because, according to every kind of mathematics, never an O leaves the matrix. The only change of the initial matrix
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>>> consists of exchanging an O and an X. Exchanging two symbols does not delete any of them.
>> In particular, you have no clue about basic logic.
>
> I have this understanding of logic: If two symbols are exchanged, then none of them is deleted.

your exchange, or swapparoo, is a step by step process, which you have admitted before does not work on infinite sets.

>
>> A In order to prove an inconsistency in ZFC
>
> it is sufficient to prove that no O can leave the matrix.

no, it is wrong for infinite sets.

>
> Regards, WM

now correct your mistakes and stop trying to mislead people.

You still get an F, for Fail.

WM <askasker48@gmail.com> writes:

> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 03:33:25 UTC+2:
>
>> I said we can get rid of the Os in WMaths without using anything from
>> Cantor, just the maths in your book.
>
> And that is clearly wrong because,

Then you will need to correct your book (again). But you won't see what
needs correcting until you follow the argument I presented based solely
on the maths in your textbook.

> according to every kind of
> mathematics, never an O leaves the matrix. The only change of the
> initial matrix
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> consists of exchanging an O and an X. Exchanging two symbols does not
> delete any of them.

And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
conveniently written as a function from N to {0,1}, has no 0s in the
image. Either you need to re-write something in your book or you should
define, more precisely what you mean by the result of an endless
sequence of swaps.

>> I could set a sequence of
>> exercises, based on your book, that would take you through the steps to
>> see how the Os "disappear" without anything from Cantor -- just WMaths.
>> Would you like me to do that?
>
> Please try it.

OK. (1) Use your k to write the initial matrix you show a bit of above
as a function from N to {0,1}. If there are terms you are not sure
about, please ask. Don't use any maths not in your book.

I can break this down a bit if it's too big an initial step.

>> >> > Or did I misinterpret Cantor?
>> >> I don't know. Do you have a citation for his discussion of this matrix?
>> >
>> > No, the citation is written above.
>> What does "no but it's above mean"?
>
> Cantor said that all fractions can be indexed. That means translated
> to my model that all O's disappear. Can you follow that thought?

OK, you don't have a citation. I didn't really think you did.

--
Ben.

Gus Gassmann schrieb am Sonntag, 4. September 2022 um 16:20:32 UTC+2:
> On Sunday, 4 September 2022 at 10:55:28 UTC-3, WM wrote:
> [...]
> > I have this understanding of logic: If two symbols are exchanged, then none of them is deleted.
> This is correct, but insufficient.

It is all that is needed.

> > > A In order to prove an inconsistency in ZFC
> >
> > it issufficient to prove that no O can leave the matrix.
> Ah, no. The fact that the reciprocal 1/n is positive for every natural number n is insufficient to conclude that the limit of the sequence {1, 1/2, 1/3, ..} cannot be 0.

The limit of a sequence is a number which is approached. The absence of all O's is not a state which is approached. The sequence ℵo, ℵo, ℵo, ... has limit ℵo, not 0.

Regards, WM

Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 18:14:07 UTC+2:
> WM <askas...@gmail.com> writes:

> > according to every kind of
> > mathematics, never an O leaves the matrix. The only change of the
> > initial matrix
> >
> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...
> >
> > consists of exchanging an O and an X. Exchanging two symbols does not
> > delete any of them.
> And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
> conveniently written as a function from N to {0,1}, has no 0s in the
> image.

You are lying.

> Either you need to re-write something in your book or you should
> define, more precisely what you mean by the result of an endless
> sequence of swaps.

The result of an endless sequence ℵo, ℵo, ℵo, ... is ℵo.

> OK. (1) Use your k to write the initial matrix you show a bit of above
> as a function from N to {0,1}. If there are terms you are not sure
> about, please ask. Don't use any maths not in your bo000ok.

I prefer to write X and O in the way I did. Nothing in my book contradicts that.

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

XXOO...
OOOO...
XOOO...
XOOO...
....

XXOO...
XOOO...
OOOO...
XOOO...
....

XXXO...
XOOO...
OOOO...
OOOO...
....
....

> > Cantor said that all fractions can be indexed. That means translated
> > to my model that all O's disappear. Can you follow that thought?
> OK, you don't have a citation. I didn't really think you did.

Here is the citation. But you need a little bit of brain to derive the above sequence of matrices.

So the set of all positive rational numbers is countable as was shown by Cantor [Cantor, p. 126. G. Cantor, letter to R. Lipschitz (19 Nov 1883)] by this sequence

1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, ...

where the sum s of numerator and denominator grows stepwise by 1, and for fixed sum the numerator grows stepwise by 1 from minimum 1 to maximum s - 1. If repeating values of fractions are eliminated, every positive rational number appears exactly once, if not, we get the sequence of all positive fractions where k = (m + n - 1)(m + n - 2)/2 + m is the index or position of fraction m/n. [Cantor, p. 132]

Regards, WM

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

> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 18:14:07 UTC+2:
>> WM <askas...@gmail.com> writes:
>
>> > according to every kind of
>> > mathematics, never an O leaves the matrix. The only change of the
>> > initial matrix
>> >
>> > XOOO...
>> > XOOO...
>> > XOOO...
>> > XOOO...
>> > ...
>> >
>> > consists of exchanging an O and an X. Exchanging two symbols does not
>> > delete any of them.
>> And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
>> conveniently written as a function from N to {0,1}, has no 0s in the
>> image.
>
> You are lying.

I showed you how. You flipped immediately back into whatever you are
calling maths that isn't yours these day. You appeared not to have
grasped even the first stage of my argument. Or maybe you did grasp it
and you decided you had to do something to avoid considering it.

>> Either you need to re-write something in your book or you should
>> define, more precisely what you mean by the result of an endless
>> sequence of swaps.
>
> The result of an endless sequence ℵo, ℵo, ℵo, ... is ℵo.

We're doing this with the maths in your book. I don't think ℵo is
defined in you textbook, is it? Anyway, that's not the result given by
WMaths as my argument showed.

>> OK. (1) Use your k to write the initial matrix you show a bit of above
>> as a function from N to {0,1}. If there are terms you are not sure
>> about, please ask. Don't use any maths not in your book.
>
> I prefer to write X and O in the way I did. Nothing in my book
> contradicts that.

Indeed. It all works in WMaths. If do decide to take me up on the
offer, and do the exercises, you'll see how it's done.

When you said "please try it" in reply to may offer of taking you
through some exercises to explain the argument, I thought you were
agreeing. But "please try it" could also be a challenge: "just try it
and see how for you get, mate"!

The offer still stands and, as I said, I can break it down (into two) if
you think that would help.

>> OK, you don't have a citation. I didn't really think you did.
>
> Here is the citation.

<rudeness cut>

> So the set of all positive rational numbers is countable as was shown
> by Cantor [Cantor, p. 126. G. Cantor, letter to R. Lipschitz (19 Nov
> 1883)] by this sequence
>
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, ...
>
> where the sum s of numerator and denominator grows stepwise by 1, and
> for fixed sum the numerator grows stepwise by 1 from minimum 1 to
> maximum s - 1. If repeating values of fractions are eliminated, every
> positive rational number appears exactly once, if not, we get the
> sequence of all positive fractions where k = (m + n - 1)(m + n - 2)/2
> + m is the index or position of fraction m/n. [Cantor, p. 132]

Right, so you are citing something other than Cantor discussing swapping
Xs and Os in a grid. I'd really have like to see how Cantor defined the
result of and endless sequence of swaps. Maybe he does in some other
work. Do you know of any?

--
Ben.

Ben Bacarisse schrieb:

> WM <askasker48@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
>> Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 18:14:07 UTC+2:
>>> WM <askas...@gmail.com> writes:
>>
>>> > according to every kind of
>>> > mathematics, never an O leaves the matrix. The only change of the
>>> > initial matrix
>>> >
>>> > XOOO...
>>> > XOOO...
>>> > XOOO...
>>> > XOOO...
>>> > ...
>>> >
>>> > consists of exchanging an O and an X. Exchanging two symbols does not
>>> > delete any of them.
>>> And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
>>> conveniently written as a function from N to {0,1}, has no 0s in the
>>> image.
>>
>> You are lying.
>
> I showed you how. You flipped immediately back into whatever you are
> calling maths that isn't yours these day. You appeared not to have
> grasped even the first stage of my argument. Or maybe you did grasp it
> and you decided you had to do something to avoid considering it.
>
>>> Either you need to re-write something in your book or you should
>>> define, more precisely what you mean by the result of an endless
>>> sequence of swaps.
>>
>> The result of an endless sequence ℵo, ℵo, ℵo, ... is ℵo.
>
> We're doing this with the maths in your book. I don't think ℵo is
> defined in you textbook, is it? Anyway, that's not the result given by
> WMaths as my argument showed.
>
>>> OK. (1) Use your k to write the initial matrix you show a bit of above
>>> as a function from N to {0,1}. If there are terms you are not sure
>>> about, please ask. Don't use any maths not in your book.
>>
>> I prefer to write X and O in the way I did. Nothing in my book
>> contradicts that.
>
> Indeed. It all works in WMaths. If do decide to take me up on the
> offer, and do the exercises, you'll see how it's done.
>
> When you said "please try it" in reply to may offer of taking you
> through some exercises to explain the argument, I thought you were
> agreeing. But "please try it" could also be a challenge: "just try it
> and see how for you get, mate"!
>
> The offer still stands and, as I said, I can break it down (into two) if
> you think that would help.
>
>>> OK, you don't have a citation. I didn't really think you did.
>>
>> Here is the citation.
>
> <rudeness cut>
>
>> So the set of all positive rational numbers is countable as was shown
>> by Cantor [Cantor, p. 126. G. Cantor, letter to R. Lipschitz (19 Nov
>> 1883)] by this sequence
>>
>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, ...
>>
>> where the sum s of numerator and denominator grows stepwise by 1, and
>> for fixed sum the numerator grows stepwise by 1 from minimum 1 to
>> maximum s - 1. If repeating values of fractions are eliminated, every
>> positive rational number appears exactly once, if not, we get the
>> sequence of all positive fractions where k = (m + n - 1)(m + n - 2)/2
>> + m is the index or position of fraction m/n. [Cantor, p. 132]
>
> Right, so you are citing something other than Cantor discussing swapping
> Xs and Os in a grid. I'd really have like to see how Cantor defined the
> result of and endless sequence of swaps. Maybe he does in some other
> work. Do you know of any?

WM does not want our math where infinity exists and he teaches this
credo to all people who are "believing" that infinity is "allowable"
which he calls dumb folks. This is WM's principal fight for many
decades now and it is the only matter in his life which he really
"posesses" as a human creature and which he is engaged in round clock.
He will never give up this "devotion" because that would also mean
to erase nearly all content in his low-IQ brain. But be sure that
he will use your great work, Ben, in a deformed shape in his further
publications. Just as an aside...

On Sunday, September 4, 2022 at 9:26:47 PM UTC+2, Ben Bacarisse wrote:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 18:14:07 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >
> >> > according to every kind of
> >> > mathematics, never an O leaves the matrix. The only change of the
> >> > initial matrix
> >> >
> >> > XOOO...
> >> > XOOO...
> >> > XOOO...
> >> > XOOO...
> >> > ...
> >> >
> >> > consists of exchanging an O and an X. Exchanging two symbols does not
> >> > delete any of them.
> >> And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
> >> conveniently written as a function from N to {0,1}, has no 0s in the
> >> image.
> >
> > You are lying.
> I showed you how. You flipped immediately back into whatever you are
> calling maths that isn't yours these day. You appeared not to have
> grasped even the first stage of my argument. Or maybe you did grasp it
> and you decided you had to do something to avoid considering it.
> >> Either you need to re-write something in your book or you should
> >> define, more precisely what you mean by the result of an endless
> >> sequence of swaps.
> >
> > The result of an endless sequence ℵo, ℵo, ℵo, .... is ℵo.
> We're doing this with the maths in your book. I don't think ℵo is
> defined in you textbook, is it? Anyway, that's not the result given by
> WMaths as my argument showed.
> >> OK. (1) Use your k to write the initial matrix you show a bit of above
> >> as a function from N to {0,1}. If there are terms you are not sure
> >> about, please ask. Don't use any maths not in your book.
> >
> > I prefer to write X and O in the way I did. Nothing in my book
> > contradicts that.
> Indeed. It all works in WMaths. If do decide to take me up on the
> offer, and do the exercises, you'll see how it's done.
>
> When you said "please try it" in reply to may offer of taking you
> through some exercises to explain the argument, I thought you were
> agreeing. But "please try it" could also be a challenge: "just try it
> and see how for you get, mate"!
>
> The offer still stands and, as I said, I can break it down (into two) if
> you think that would help.
> >> OK, you don't have a citation. I didn't really think you did.
> >
> > Here is the citation.
> <rudeness cut>
> > So the set of all positive rational numbers is countable as was shown
> > by Cantor [Cantor, p. 126. G. Cantor, letter to R. Lipschitz (19 Nov
> > 1883)] by this sequence
> >
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, ...
> >
> > where the sum s of numerator and denominator grows stepwise by 1, and
> > for fixed sum the numerator grows stepwise by 1 from minimum 1 to
> > maximum s - 1. If repeating values of fractions are eliminated, every
> > positive rational number appears exactly once, if not, we get the
> > sequence of all positive fractions where k = (m + n - 1)(m + n - 2)/2
> > + m is the index or position of fraction m/n. [Cantor, p. 132]
> Right, so you are citing something other than Cantor discussing swapping
> Xs and Os in a grid. I'd really have like to see how Cantor defined the
> result of and endless sequence of swaps. Maybe he does in some other
> work. Do you know of any?
>
> --
> Ben.

There is only one immutable rule in what you call WMaths and I call Muckmeatics:
"You cannot nail a pudding to the wall."
Once you know that you know everything worth knowing about this subject,
which is best described as pseudo-mathematical polemics; or, of you prefer,
polemical pseudo-mathematics.

Ben Bacarisse schrieb am Sonntag, 4. September 2022 um 21:26:47 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 Sonntag, 4. September 2022 um 18:14:07 UTC+2:
> >> WM <askas...@gmail.com> writes:
> >
> >> > according to every kind of
> >> > mathematics, never an O leaves the matrix. The only change of the
> >> > initial matrix
> >> >
> >> > XOOO...
> >> > XOOO...
> >> > XOOO...
> >> > XOOO...
> >> > ...
> >> >
> >> > consists of exchanging an O and an X. Exchanging two symbols does not
> >> > delete any of them.
> >> And yet, in WMaths, the result of an endless sequence of swaps is a matrix,
> >> conveniently written as a function from N to {0,1}, has no 0s in the
> >> image.
> >
> > You are lying.
> I showed you how.

You cannot show it because it is wrong. It is nonsense to believe that by exchanging two items, one of them disappears.

> You flipped immediately back into whatever you are
> calling maths that isn't yours these day. You appeared not to have
> grasped even the first stage of my argument.

Your argument started with a wrong image of Cantor's matrix which I corrected.

> > The result of an endless sequence ℵo, ℵo, ℵo, .... is ℵo.
> We're doing this with the maths in your book. I don't think ℵo is
> defined in you textbook, is it?

There is no actual infinity in my book. There is no 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, ...
....

But we have to assume it in order to discuss it by means of correct mathematics from hatever books.

> Anyway, that's not the result given by
> WMaths as my argument showed.

Your argument failed to produce exchanges.

> > I prefer to write X and O in the way I did. Nothing in my book
> > contradicts that.
> Indeed. It all works in WMaths. If do decide to take me up on the
> offer, and do the exercises, you'll see how it's done.
>
> When you said "please try it" in reply to may offer of taking you
> through some exercises

I am looking foreward to your yexplaination of the argument. It deals with X and O starting from the matrix

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

not with a linear sequence.

> The offer still stands and, as I said, I can break it down (into two) if
> you think that would help.

Show what you have, but adhere to the topic and don't try to change it..

> >> OK, you don't have a citation. I didn't really think you did.
> >
> > Here is the citation.
> <rudeness cut>

No rudeness. Only a necessary hint that it requires some thinking to understand the language of matrices I use. Many are unable.

> > So the set of all positive rational numbers is countable as was shown
> > by Cantor [Cantor, p. 126. G. Cantor, letter to R. Lipschitz (19 Nov
> > 1883)] by this sequence
> >
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, ...
> >
> > where the sum s of numerator and denominator grows stepwise by 1, and
> > for fixed sum the numerator grows stepwise by 1 from minimum 1 to
> > maximum s - 1. If repeating values of fractions are eliminated, every
> > positive rational number appears exactly once, if not, we get the
> > sequence of all positive fractions where k = (m + n - 1)(m + n - 2)/2
> > + m is the index or position of fraction m/n. [Cantor, p. 132]

> Right, so you are citing something other than Cantor discussing swapping
> Xs and Os in a grid.

I warned you. Mathematics is interweaved with thinking.

> I'd really have like to see how Cantor defined the
> result of and endless sequence of swaps. Maybe he does in some other
> work. Do you know of any?

If he had tried it, he would have seen the result.

Cantor used the natural numbers for indexing, I use the integer fractions. That is the only new idea. Since the integer fractions also have to be indexed, we get the swaps.

Regards, WM

Tom Bola schrieb am Sonntag, 4. September 2022 um 22:14:09 UTC+2:

> WM does not want our math where infinity exists and he teaches this
> credo to all people who are "believing" that infinity is "allowable"
> which he calls dumb folks

Wrong. I accept and assume the actually infinite 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, ...
....

Regards, WM

JVR schrieb am Sonntag, 4. September 2022 um 22:14:39 UTC+2:
>
> "You cannot nail a pudding to the wall."

You are the pudding! You claimed you could explain how the O's were removed from the matrix

XOO...
XOO...
XOO...
....

by simple exchanges with the X's.

When you are challenged to show it, either you escape to insults or poems.

Regards, WM

WM drivels:

>> WM does not want our math where infinity exists and he teaches this
>> credo to all people who are "believing" that infinity is "allowable"
>> which he calls dumb folks
>
> Wrong. I accept and assume the actually infinite 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, ...
> ...

Which is endless and: |IN| = |IN x IN|
You idiotic asshole do not WANT our infinity.

Until you accept Hilbert's Hotel it makes no sense to reply your drivel,
and you psychotic never will accept that.

Piss off, clown.