For corrections: Proof of the existence of dark numbers

[WM]

Abstract: We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.


1. Outline of the proof

(1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.
(2) Then we distribute, according to Cantor's prescription, these indices over the whole matrix. We observe that in every step prescribed by Cantor the set of indices does not increase and the set of not indexed fractions does not decrease.
(3) Therefore it is impossible to index all fractions in a definable way. Indexing many fractions "in the limit" would be undefined and can be excluded according to section 2 below.
(4) After having issued all indices only indexed fractions can be seen in the matrix.
(5) We conclude from the existing but invisible not indexed fractions that they are sitting at invisible positions, inside of the matrix.
(6) Hence also the first column of the matrix and therefore also  has invisible, co-called dark elements.
(7) Hence also the initial mapping of natural numbers and integer fractions cannot have been complete. Bijections, i.e., complete mappings, of infinite sets and are impossible.


2. Rejecting the limit idea

When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.

"If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed place of a simple infinite sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126]

"The infinite sequence thus defined has the peculiar property to contain all positive rational numbers and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one from this epitome () has been forgotten." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]

"such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

The clearness of these expressions is noteworthy: all and every, at an absolutely fixed place, at a definite position, not a single one has been forgotten.

"In fact, according to the above definition of cardinality, the cardinal number M remains unchanged if in place of an element or of each of some elements, or even of each of all elements m of M another thing is substituted." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 283]

This fact will be utilized to replace the pairs of the bijection by matrices or to attach a matrix to every pair of the bijection, respectively.


3. The proof

If all positive fractions m/n are existing, then they all are contained in 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, ...
    

If all natural numbers k are existing, then they can be used as indices to index the integer fractions m/1 of the first column. Denoting indexed fractions by X and not indexed fractions by O, we obtain the following matrix:

XOOO...
XOOO...
XOOO...
XOOO...
XOOO...
    

Cantor claimed that all natural numbers k are existing and can be applied to index all positive fractions m/n. They are distributed according to

k = (m + n - 1)(m + n - 2)/2 + m.
The result is a sequence of fractions

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

This sequence is modelled here in the language of matrices. The indices are taken from their initial positions in the first column and are distributed in the given order.

Index 1 remains at the first term 1/1. The next term 1/2 is indexed with 2 which is taken from position 2/1.

XXOO...
OOOO...
XOOO...
XOOO...
XOOO...
    

Then index 3 it taken from 3/1 and attached to 2/1:

XXOO...
OOOO...
XOOO...
XOOO...
XOOO...
    

Then index 4 it taken from 4/1 and attached to 1/3:

XXXO...
XOOO...
OOOO...
OOOO...
XOOO...
    

Then index 5 it taken from 5/1 and attached to 2/2:

XXXO...
XXOO...
OOOO...
OOOO...
OOOO...
    

And so on. When finally all exchanges of X and O have been carried through and, according to Cantor, all indices have been issued, it turns out that no fraction without index is visible any longer

XXXX...
XXXX...
XXXX...
XXXX...
XXXX...
    

but by the technique of lossless exchange of X and O no O can have left the matrix as long as finite natural numbers are issued as indices. Therefore there are not less fractions without index than at the beginning.

We know that all O and as many fractions without index are remaining, but we cannot find any one. Where are they? Fractions that cannot be found we call dark. The O stay at dark positions. This is the only explanation.

This proof shows that every column has dark positions. Therefore also the integer fractions and also the natural numbers contain dark elements. Cantor's enumerating concerns only the visible fractions, not all fractions. This concerns also every other attempt to enumerate the fractions and even the identical mapping. Bijections, i.e., complete mappings, of infinite sets and |N are impossible.


4. Counterarguments

Sometimes we hear the argument that, in spite of the preconditions explicitly quoted in section 2, a set-theoretical or analytical*) limit should be applied. This however would imply that all the O remain present in all defined matrices until "in the limit" these infinitely many O have to leave in an undefinable way; hence infinitely many fractions have to become indexed such that none of them can be checked - contrary to the proper meaning of indexing.

Some set theorists reject it as inadmissible to limit the indices by starting in the first column. But that means only to check that the set of natural numbers has same size as the set of integer fractions.

Others would tolerate that lossless exchange of X and O can suffer from losses. That argument can be excluded by basic logic.

*) Note that an analytical limit like 0 is approached by the sequence (1/n) but never attained. A bijective mapping of sets however is claimed to be completed, according to section 2.

Regards, WM

[Ben Bacarisse]

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

> When finally all exchanges of X and O have been carried through

A notion (the final result of an endless sequence of swaps) that you
refuse to define or explain, despite it being at the heart of what you
present as a mystery. You reject the obvious meaning (the limit of a
sequence of index functions) because then WMaths gives the result you
consider to be "wrong"!

> and, according to Cantor,

And you also can't give a reference showing how Cantor defines or
explains this key notion ether. That's because, if your failure to cite
a source is any indication, he never did define what the final result of
a sequence of swaps is.

(I'm not sure I do trust your failure to cite though, since (a) you are
not a reliable historian and (b) Cantor would surely have known how to
define it.)

--
Ben.

[Sergi o]

On 11/16/2022 12:59 PM, WM wrote:
> Abstract: We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
>
>
> 1. Outline of the proof
>
> (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.

corrected: in a one to one mapping.

We are done. nothing else is needed.

What follows is a poor attempt to disprove Cantor, (although (1) accepts Cantor) by obscure, using O pasties, and X stickies *replacing fractions*,
and overwriting the matrix of rationals with pasties and stickies in a step by step process in an infinite set.

> (2) Then we distribute, according to Cantor's prescription, these indices over the whole matrix. We observe that in every step prescribed by Cantor the set of indices does not increase and the set of not indexed fractions does not decrease.
> (3) Therefore it is impossible to index all fractions in a definable way. Indexing many fractions "in the limit" would be undefined and can be excluded according to section 2 below.
> (4) After having issued all indices only indexed fractions can be seen in the matrix.
> (5) We conclude from the existing but invisible not indexed fractions that they are sitting at invisible positions, inside of the matrix.
> (6) Hence also the first column of the matrix and therefore also  has invisible, co-called dark elements.
> (7) Hence also the initial mapping of natural numbers and integer fractions cannot have been complete. Bijections, i.e., complete mappings, of infinite sets and are impossible.
>
>
> 2. Rejecting the limit idea
>
> When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.

wrong, there never was/is a limit.

<snip crap>

>
> Regards, WM

[WM]

Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > When finally all exchanges of X and O have been carried through
> A notion (the final result of an endless sequence of swaps) that you
> refuse to define or explain, despite it being at the heart of what you
> present as a mystery.

See section 2. There the final state is defined by Cantor. Further every set theorist should know that, if a bijection exists, all indices must have been applied. My matrices do not repeat this till the final state because that is not necessary for my proof.

My argument is this: If all fractions get indexed, each one by one and only one index, then all O's must leave the matrix in an ordered way. There are infinitely many O's. Therefore there must be a first O leaving at a finite step. But this can be excluded for every finite step. Therefore the O's do not leave the matrix or they don't leave at finite steps. In both cases there is no bijection.

My sequence of matrices does not need to reach an end but only every finite step. That cannot be forbidden.

> You reject the obvious meaning (the limit of a
> sequence of index functions) because then WMaths gives the result you
> consider to be "wrong"!

I do not reject the limit but I know that every fraction must get indexed before any limit.
>
> > and, according to Cantor,

> (I'm not sure I do trust your failure to cite though, since (a) you are
> not a reliable historian and (b) Cantor would surely have known how to
> define it.)

Did you read section 2? There it is defined: "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed place of a simple infinite sequence". "Werden nun die Zahlen p/q in einer solchen Reihenfolge gedacht, [...] so kommt jede Zahl p/q an eine ganz bestimmte Stelle einer einfach unendlichen Reihe,"

This sequence must exist. I do not need the end but only prove that at all finite steps no O leaves the matrix. Therefore they don't leave at all or they leave in your "limit" all together, excluding the requirement to be indexed in a distinguishable way "at an absolutely fixed place".

Regards, WM

[WM]

Sergi o schrieb am Mittwoch, 16. November 2022 um 23:50:12 UTC+1:
> On 11/16/2022 12:59 PM, WM wrote:

> > (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.
> corrected: in a one to one mapping.
>
> We are done. nothing else is needed.

The proof is needed.

>
> What follows is a poor attempt to disprove Cantor, (although (1) accepts Cantor) by obscure, using O pasties, and X stickies *replacing fractions*,

Try to get it: A not indexed fraction is represented by an O. If all fractions get indexed, then all O's must leave the matrix.

> and overwriting the matrix of rationals with pasties and stickies in a step by step process in an infinite set.

My argument covers only finite steps. That is allowed if a sequence exists. If all fractions were indexed, then the infinitely many O's had to leave the matrix in an ordered way, i.e., a first one had to leave at a finite step - before any "limit" or "end".

This however is impossible.

> > 2. Rejecting the limit idea
> >
> > When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.
> wrong, there never was/is a limit.

That is true for Cantor, but see BB's argument.

Regards, WM

[Ben Bacarisse]

WM <askas...@gmail.com> writes:

> Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>> > When finally all exchanges of X and O have been carried through
>> A notion (the final result of an endless sequence of swaps) that you
>> refuse to define or explain, despite it being at the heart of what you
>> present as a mystery.
>
> See section 2. There the final state is defined by Cantor.

No. You give a few un-contentious quotes, none of which suggest that
Cantor ever discussed the final result of an endless sequence of swaps.
You don't define it, and neither did Cantor. It's the key mystery you
are relying on here. Define it, and you are lost.

--
Ben.

[Sergi o]

On 11/17/2022 3:42 AM, WM wrote:
> Sergi o schrieb am Mittwoch, 16. November 2022 um 23:50:12 UTC+1:
>> On 11/16/2022 12:59 PM, WM wrote:
>
>>> (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.
>> corrected: in a one to one mapping.
>>
>> We are done. nothing else is needed.
>
> The proof is needed.

Proof is already done by Cantor, and you accepted that in (1).

besides, you dont do proofs, you dont read or understand them.

>>
>> What follows is a poor attempt to disprove Cantor, (although (1) accepts Cantor) by obscure, using O pasties, and X stickies *replacing fractions*,
>
> Try to get it: A not indexed fraction is represented by an O. If all fractions get indexed, then all O's must leave the matrix.

so you admit you replaced indexed fraction with an O. => because you need a smoke screen for your deception.

Cantor just use the fractions as is, and proved his enumeration.

>
>> and overwriting the matrix of rationals with pasties and stickies in a step by step process in an infinite set.
>
> My argument covers only finite steps.

"finite steps" cannot be applied to both infinite sets involved here. Fail.

> That is allowed if a sequence exists. If all fractions were indexed, then the infinitely many O's had to leave the matrix in an ordered way, i.e., a first one had to leave at a finite step - before any "limit" or "end".
>
> This however is impossible.

only because you stopped.

>
>>> 2. Rejecting the limit idea
>>>
>>> When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.
>> wrong, there never was/is a limit.
>
> That is true for Cantor, but see BB's argument.

you do not understand anything of what is being talked about here. You are out of Math.

>
> Regards, WM

[WM]

Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 14:32:02 UTC+1:
> WM <askas...@gmail.com> writes:
>
> > Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
> >> WM <askas...@gmail.com> writes:
> >> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> >> Unendlichen" at Hochschule Augsburg.)
> >> > When finally all exchanges of X and O have been carried through
> >> A notion (the final result of an endless sequence of swaps) that you
> >> refuse to define or explain, despite it being at the heart of what you
> >> present as a mystery.
> >
> > See section 2. There the final state is defined by Cantor.
> No. You give a few un-contentious quotes, none of which suggest that
> Cantor ever discussed the final result of an endless sequence of swaps.

You have misunderstood.

Cantor did not discuss swaps but constructed his sequence. He claimed that *his* construction (often he talks about his "process") has a final result, namely the complete enumeration of fractions.

I repeat his process, i.e., I follow his sequence by my matrices as far as only finite indices are concerned. My finding is that never a O disappears. That's all and that's enough to exclude a complete enumeration by distinguishable numbers.

Regards, WM

[Sergi o]

On 11/17/2022 8:00 AM, WM wrote:
> Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 14:32:02 UTC+1:
>> WM <askas...@gmail.com> writes:
>>
>>> Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
>>>> WM <askas...@gmail.com> writes:
>>>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>>>> Unendlichen" at Hochschule Augsburg.)
>>>>> When finally all exchanges of X and O have been carried through
>>>> A notion (the final result of an endless sequence of swaps) that you
>>>> refuse to define or explain, despite it being at the heart of what you
>>>> present as a mystery.
>>>
>>> See section 2. There the final state is defined by Cantor.
>> No. You give a few un-contentious quotes, none of which suggest that
>> Cantor ever discussed the final result of an endless sequence of swaps.
>
> You have misunderstood.
>
> Cantor did not discuss swaps but constructed his sequence. He claimed that *his* construction (often he talks about his "process") has a final result, namely the complete enumeration of fractions.
>
> I repeat his process,

Liar.


>
> Regards, WM
>

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 14:32:02 UTC+1:
>> WM <askas...@gmail.com> writes:
>>
>> > Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
>> >> WM <askas...@gmail.com> writes:
>> >> > When finally all exchanges of X and O have been carried through
>> >> A notion (the final result of an endless sequence of swaps) that you
>> >> refuse to define or explain, despite it being at the heart of what you
>> >> present as a mystery.
>> >
>> > See section 2. There the final state is defined by Cantor.
>> No. You give a few un-contentious quotes, none of which suggest that
>> Cantor ever discussed the final result of an endless sequence of swaps.
>
> You have misunderstood.

I understand that you have not defined what you mean by the final result

of an endless sequence of swaps.

> Cantor did not discuss swaps

I'll take your word for it. So what do you mean when you talk about the
final result of an endless sequence of swaps? You can't claim to mean
what Cantor means because, apparently, Canter did not even /discuss/
swaps, much less define the meaning of your magic phrase.

Go on, define it.

And if you can't (I know you can't) at least calculate for us one
example. Given the sequence s(n) = n and the sequence of swaps x(n) =
(n, 2n). Tell us what /you/ mean when you talk about the "final result"
of applying the x(n) to s. Maybe I can help you define it.

--
Ben.

[WM]

Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 18:42:09 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 14:32:02 UTC+1:
> >> WM <askas...@gmail.com> writes:
> >>
> >> > Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
> >> >> WM <askas...@gmail.com> writes:
> >> >> > When finally all exchanges of X and O have been carried through
> >> >> A notion (the final result of an endless sequence of swaps) that you
> >> >> refuse to define or explain, despite it being at the heart of what you
> >> >> present as a mystery.
> >> >
> >> > See section 2. There the final state is defined by Cantor.
> >> No. You give a few un-contentious quotes, none of which suggest that
> >> Cantor ever discussed the final result of an endless sequence of swaps.
> >
> > You have misunderstood.
> I understand that you have not defined what you mean by the final result
> of an endless sequence of swaps.
> > Cantor did not discuss swaps
> I'll take your word for it.

In section 2 you can find Cantor's essential statements. He did not know my approach and did not comment it.

> So what do you mean when you talk about the
> final result of an endless sequence of swaps?

I do not talk about that result. I use Cantor's result, the complete sequence of fractions where none is mising. My sequence of swaps is only necessary up to every finite term.

> And if you can't (I know you can't)

I hope that after a while you will understand the argument outlined in section 1.
Cantor did not talk about swaps but about a complete sequence. That is shown in section 2.
I repeat this sequence by matrices up tho every finite term. That is shown in section 3.

If all fractions are indexed by distinguishable indices, then the first O must leave the matrix at a finite step reached by my sequence of matrices (because afterwards infinitely many further O's must leave at different steps). This is disproved.

> at least calculate for us one
> example. Given the sequence s(n) = n and the sequence of swaps x(n) =
> (n, 2n). Tell us what /you/ mean when you talk about the "final result"
> of applying the x(n) to s. Maybe I can help you define it.

I have shown that Cantor's claimed final result does not exist in a definable way. Why should your task have a final result? Why should I talk about it?

Regards, WM

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 18:42:09 UTC+1:
>> WM <askas...@gmail.com> writes:
>>
>> > Ben Bacarisse schrieb am Donnerstag, 17. November 2022 um 14:32:02 UTC+1:
>> >> WM <askas...@gmail.com> writes:
>> >>
>> >> > Ben Bacarisse schrieb am Mittwoch, 16. November 2022 um 21:31:24 UTC+1:
>> >> >> WM <askas...@gmail.com> writes:
>> >> >> > When finally all exchanges of X and O have been carried through
>> >> >> A notion (the final result of an endless sequence of swaps) that you
>> >> >> refuse to define or explain, despite it being at the heart of what you
>> >> >> present as a mystery.
>> >> >
>> >> > See section 2. There the final state is defined by Cantor.
>> >> No. You give a few un-contentious quotes, none of which suggest that
>> >> Cantor ever discussed the final result of an endless sequence of swaps.
>> >
>> > You have misunderstood.
>> I understand that you have not defined what you mean by the final result
>> of an endless sequence of swaps.
>> > Cantor did not discuss swaps

>> So what do you mean when you talk about the
>> final result of an endless sequence of swaps?
>
> I do not talk about that result.

Don't quibble. You present the state of affairs "when all exchanges
have been carried out" but you don't define what it means to carry out
all the exchanges.

> I use Cantor's result,

No you can't because, as you admit, Cantor does not even talk about
swaps, much less define what the state of affairs would be "when all
exchanges have been carried out".

>> And if you can't (I know you can't)
>

> I hope that after a while you will understand ...

As expected, no definition of what you meant by "when all exchanges have
been carried out".

> I have shown that Cantor's claimed final result

Cantor, as you admit, does not define the state of affairs "when all
exchanges have been carried out" because he does not even discuss swaps.

Basically you present a result, based on the currently unspecified
notion of "when all exchanges have been carried out", with no
justification because you can't say what that magic phrase means.

Well, you can, of course, because I showed you a reasonable definition
of what it means, but you didnn't like it because it works in WMaths as
well!

--
Ben.

[Fritz Feldhase]

On Friday, November 18, 2022 at 2:41:21 AM UTC+1, Ben Bacarisse wrote:

> Basically you present a result, based on the currently unspecified
> notion of "when all exchanges have been carried out", with no
> justification because you can't say what that magic phrase means.

Hint: ""[WM’s] conclusions are based on the sloppiness of his notions, his inability of giving precise definitions, his fundamental misunderstanding of elementary mathematical concepts, and sometimes, as the late Dik Winter remarked [...], on nothing at all." " (Dr. Dr. hc Franz Lemmermeyer)

Lemmermeyer?

See: https://www.amazon.de/B%C3%BCcher-Franz-Lemmermeyer/s?rh=n%3A186606%2Cp_27%3AFranz+Lemmermeyer

[WM]

Ben Bacarisse schrieb am Freitag, 18. November 2022 um 02:41:21 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)

> > I use Cantor's result,
> No you can't because, as you admit, Cantor does not even talk about
> swaps, much less define what the state of affairs would be "when all
> exchanges have been carried out".

You don't wish to understand? Looking for an escape? There is none!
Cantor claims that *his* method or process yields a final result: Every natural number is an index of fraction and every fraction has a natural index. In modern terminology: Bijection.

>
> Basically you present a result, based on the currently unspecified
> notion of "when all exchanges have been carried out", with no
> justification because you can't say what that magic phrase means.

This same final result would be accomplished by my sequence of matrices, because it is only another language for the same facts.

But I do not need this result (and therefore I do not need to specify it better than Cantor specified his) because I need only prove that in no finite step any O leaves the matrix. If Cantor's final result would be correct that every fraction has an index (and not only every index has a fraction) then the O's would leave the matrix - and a first O would leave at some final step which is reached with no qualms by my sequence of matrices.

Regards, WM

[WM]

Fritz Feldhase schrieb am Freitag, 18. November 2022 um 03:21:59 UTC+1:

> Hint:

This is the topic:

Cantor claims that *his* method or process yields a final result: Every natural number is an index of a fraction and every fraction has a natural index. In modern terminology: Bijection.

I prove that in no finite step any O leaves the matrix. Infinitely many fractions remain without index. They must get indexed after all finite steps or never. That excludes definable indexing. Cantors claim "so kommt jede Zahl p/q an eine ganz bestimmte Stelle einer einfach unendlichen Reihe" is disproved, denn alle ganz bestimmten Stellen werden von meiner Matrixfolge erreicht.

Regards, WM

[WM]

Ben Bacarisse schrieb am Freitag, 18. November 2022 um 02:41:21 UTC+1:

> Basically you present a result, based on the currently unspecified
> notion of "when all exchanges have been carried out", with no
> justification because you can't say what that magic phrase means.

After all this is a good point. It was worthwhile to open this thread "For corrections". I have changed the wording of my argument
from

(4) After having issued all indices only indexed fractions can be seen in the matrix.

into
(4) In case of a complete mapping of |N into the matrix, i.e., when every index has entered its final position, only indexed fractions are visible in the matrix.

All exchanges have been carried out means the same as all indices have been applied to fractions. This process is claimed to be completed by Cantor. This process can be claimed to be completed by myself. Because: What is the difference?

Cantor takes natural numbers from wherever and attaches them to fractions.
I take natural numbers from the first column and attach them to fractions. The only difference is: when I take a number, then I remember that it is no longer at its original place. This is indicated by an O. Does this small difference make my method invalid? Hardly. It only shows that Cantor's method is invalid.

But as I told you already, I need no final state. All finite steps are sufficient to exclude a complete mapping.

Regards, WM

[Ben Bacarisse]

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

> This is the topic:
>
> Cantor claims that *his* method or process yields a final result:

No he does not, and you know it. You have clearly stated that Cantor
does not even discuss swaps, much less the "final result" that you
present as if it were Cantor's. Of course, as I showed, you can indeed
define the "final result", but you don't like my definition because it
gives the mysterious result in WMaths as well as in proper mathematics.

But you must avoid saying what you mean at all costs while pretending
the result is someone else's fault.

--
Ben.

[Sergi o]

On 11/18/2022 8:15 AM, WM wrote:
> Ben Bacarisse schrieb am Freitag, 18. November 2022 um 02:41:21 UTC+1:
>
>> Basically you present a result, based on the currently unspecified
>> notion of "when all exchanges have been carried out", with no
>> justification because you can't say what that magic phrase means.
>

<snip crap>

>


> Cantor takes natural numbers from wherever

no, Cantor took them out of bucket.

> and attaches them to fractions.

Yes, Cantor used pasties to attach or stick, a natural number onto each one of the willing fraction.

> I take natural numbers from the first column and attach them to fractions.

see, you used stickies too, to glue, attach, each number from the bucket onto, and covering up the poor suffocating fraction.

>The only difference is: when I take a number, then I remember that it is no longer at its original place.

Shameful!! you ripped out fractions in the first column, and used glue to pastie each onto strange and unwilling fractions!

> This is indicated by an O.

No! This is indicated by a BOZO, who runs amuck putting stickies and pasties on everything and canto keep track of any of it.

>Does this small difference make my method invalid?

does fake math poo stink ?

> Hardly. It only shows that Cantor's method is invalid.
>
> But as I told you already, I need no final state. All finite steps are sufficient to exclude a complete mapping.

you are permanently in your final state of confusion,

your brain has been sufficiently excluded for a complete mapping,

however you can donate it to medical science at https://www.youtube.com/watch?v=mr2kIROP00Y

>
> Regards, WM
>
>
>
>
>
>
>

[WM]

Ben Bacarisse schrieb am Samstag, 19. November 2022 um 02:36:14 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > This is the topic:
> >
> > Cantor claims that *his* method or process yields a final result:
> No he does not, and you know it.

You are a liar and I can prove it:

"then every number p/q comes at an absolutely fixed place of a simple infinite sequence"

"The infinite sequence thus defined has the peculiar property to contain all positive rational numbers and each of them only once at a determined place."

"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one of this epitome () has been forgotten."

> You have clearly stated that Cantor
> does not even discuss swaps,

Cantor discusses counting, i.e., attaching indices to elements like fractions (or algebraic numbers). That means taking an index from |N and attaching it to the element to be counted.
I do the same. The only difference is that I remember the origin of the indices and note that its origin now is empty. He does not.

> much less the "final result" that you present as if it were Cantor's.

"all positive rational numbers and each of them only once at a determined place" That is the final result.

> Of course, as I showed, you can indeed
> define the "final result", but you don't like my definition

Since you are a liar, your definition presumably will also be a lie. I have defined the final result in accordance with Cantor: Every index X has attained its final place.

> But you must avoid saying what you mean

You are lying again. The final result ist that the process has been finished and nothing is happening any more.

> the result is someone else's fault.

As I have shown, I do not even need the final result because it is sufficient to prove that in every finite step no O will disappear.

Regards, WM

[Sergi o]

On 11/19/2022 6:07 AM, WM wrote:
> Ben Bacarisse schrieb am Samstag, 19. November 2022 um 02:36:14 UTC+1:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>> This is the topic:
>>>
>>> Cantor claims that *his* method or process yields a final result:
>> No he does not, and you know it.
>
> You are a liar and I can prove it:
> "then every number p/q comes at an absolutely fixed place of a simple infinite sequence"
> "The infinite sequence thus defined has the peculiar property to contain all positive rational numbers and each of them only once at a determined place."
> "thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one of this epitome () has been forgotten."
>

none of that applies.

>> You have clearly stated that Cantor
>> does not even discuss swaps,
>
> Cantor discusses counting, i.e., attaching indices to elements like fractions (or algebraic numbers). That means taking an index from |N and attaching it to the element to be counted.

its called one to one mapping, bijection, it is NOT SWAPPING liar.

> I do the same.

Liar! you do nothing of the sort, no equations, no math from WM.

> The only difference is that I remember the origin of the indices and note that its origin now is empty.

Liar, you have swaps, switcharoos, stickies and pasties, cluster f*ck, all based upon NO MATH

>
>> much less the "final result" that you present as if it were Cantor's.
>
> "all positive rational numbers and each of them only once at a determined place" That is the final result.

Liar, you dont know what got swapped, your stickies and pasties are mixed up, like where did your Os go ?

>
>> Of course, as I showed, you can indeed
>> define the "final result", but you don't like my definition
>
> Since you are a liar, your definition presumably will also be a lie. I have defined the final result in accordance with Cantor: Every index X has attained its final place.

no, you say you disprove Cantor, and you say you are in accordance with Cantor. and from you NO MATH.

>
>> But you must avoid saying what you mean
>
> You are lying again. The final result ist that the process has been finished and nothing is happening any more.

you have no process, swappadoodlefestavale is your incoherent process of stickies and pasties.

>
>> the result is someone else's fault.
>
> As I have shown, I do not even need the final result because it is sufficient to prove that in every finite step no O will disappear.

you have shown you are in your final state of confusion.

Why dont you use equations to describe your matrix switcharoo ?? because it doesnt work.

>
> Regards, WM

[WM]

Sergi o schrieb am Samstag, 19. November 2022 um 16:10:45 UTC+1:

> Why dont you use equations to describe your matrix switcharoo ??

I do. Every X gets from k/1 to m/n where k = (m + n - 1)(m + n - 2)/2 + m while the O takes the opposite way.

Regards, WM

[Ross A. Finlayson]

Eat it, Mongol goon.

[Ben Bacarisse]

Sergi o <inv...@invalid.com> writes:

> Why dont you use equations to describe your matrix switcharoo ??
> because it doesnt work.

It does work. It not hard to turn this all into real hard maths. All
that's needed is to say what the effect of an infinite sequence of swaps
is and there is a perfectly reasonable definition for that: the limit of
a convergent sequence of functions.

The problem is that the details work in WMaths just as they do in proper
mathematics so WM has to reject that definition. So what does he mean
by the end result or the completed swaps or whatever the phrase is
today? He can't say because he can't think of any definition except
the sane one that gives the result he hates.

Some detail... There's no need to mess with matrices. For example,
consider the trivial sequence

S_0(n) = n (i.e. 1, 2, 3, ...)

Swapping (1, 2) produces

S_1(n) = 2, 1, 3, 4, 5, 6, ....

Applying the swap (2, 4) to S_1 produces

S_2(n) = 2, 4, 3, 1, 5, 6, ...

Each swap of i with 2i yields a sequence with another even number in the
initial portion, and no later swap will change the initial segment.

So what is the effect of applying the infinite sequence of swaps (i, 2i)
to S_0? The natural definition is that it's E(n) = 2n -- the sequence
of even numbers. This is simply the limit of a point-wise convergent
sequence of functions S_k.

Just as with WMs matrix, no swap changes "removes" and odd numbers (all
the S_k have image set N) but the limit sequence has no even numbers in
it!

What WM is doing is presenting the mystery ("no swap removes an X, but
the end result has no Xs") without defining what "the end result" means
because he own book explains how to fund the point-wise limit of the
sequence of matrices and he can't think of a meaning for the phrase that
would let WMaths off the hook.

--
Ben.

[WM]

Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 01:03:22 UTC+1:
> Sergi o <inv...@invalid.com> writes:
>
> > Why dont you use equations to describe your matrix switcharoo ??
> > because it doesnt work.
> It does work.

Of course. Every X gets from k/1 to m/n where k = (m + n - 1)(m + n - 2)/2 + m while the O takes the opposite way.

> All
> that's needed is to say what the effect of an infinite sequence of swaps
> is and there is a perfectly reasonable definition for that: the limit of
> a convergent sequence of functions.

The "limit" can be excluded because it would imply that all the O remain present in all definable matrices until "in the limit" these infinitely many O have to leave panic-stricken in an undefinable way; hence infinitely many fractions have to become indexed "in the limit" such that none of them can be checked - contrary to the proper meaning of indexing.

Limits are not suitable. Note that an analytical limit like 0 is approached by the sequence (1/n) but never attained. A bijective mapping of sets however must be complete, according to Cantor: The clarity of his words is noteworthy: all and every, completely, at an absolutely fixed position, νth number, where not a single one has been forgotten.

>
> The problem is that the details work in WMaths just as they do in proper
> mathematics so WM has to reject that definition. So what does he mean
> by the end result or the completed swaps or whatever the phrase is
> today? He can't say because he can't think of any definition except
> the sane one that gives the result he hates.

The definition has been given by Cantor: every number p/q comes at an absolutely fixed place of a simple infinite sequence". No limit can accomplish that.

The result is: The mapping of ℕ into the matrix is completed. Every index has entered its final position with no exception.

>
> Some detail... There's no need to mess with matrices. For example,
> consider the trivial sequence
>
> S_0(n) = n (i.e. 1, 2, 3, ...)
>
> Swapping (1, 2) produces
>
> S_1(n) = 2, 1, 3, 4, 5, 6, ....
>
> Applying the swap (2, 4) to S_1 produces
>
> S_2(n) = 2, 4, 3, 1, 5, 6, ...
>
> Each swap of i with 2i yields a sequence with another even number in the
> initial portion, and no later swap will change the initial segment.
>
> So what is the effect of applying the infinite sequence of swaps (i, 2i)
> to S_0? The natural definition is that it's E(n) = 2n -- the sequence
> of even numbers. This is simply the limit of a point-wise convergent
> sequence of functions S_k.

Thank you for this nice example. Another proof of dark numbers: The places where the odd numbers end up are dark. Or would you claim that odd numbers are deleted by simply exchanging them? They can and will not leave the sequence by swaps. But in the final state they settle at dark places.

> What WM is doing is presenting the mystery ("no swap removes an X, but
> the end result has no Xs") without defining what "the end result" means

The final state is reached when every X has settled at its position and no X further moves.
Same is with your sequence of swaps. All happens in accordance with logic. The loss of numbers from the matrix or from your sequence by swaps would suffer from deliberately contradicting basic logic.

Regards, WM

[Fritz Feldhase]

On Sunday, November 20, 2022 at 9:28:15 AM UTC+1, WM wrote:

Your BS

> suffer[s] from deliberately contradicting basic logic.

[WM]

In lossless exchange of elements, elements will get lost? Here is the wonderful and simple example:

1, 2, 3, 4, 5, 6, 7, ...
2, 1, 3, 4, 5, 6, 7, ...
2, 4, 1, 3, 5, 6, 7, ...
2, 4, 6, 1, 3, 5, 7, ...
and so on.

In every term of the sequence all odd numbers are present and are indexed.
But "in the limit" they have gone, leaving no traces?

That would contradict "that every element of the set stands at a definite position of this sequence" and of course basic logic. Note that the number of indices does never change. Obviously there are enough.

Regards, WM

[JVR]

In fact, it happens quite often that students in Analysis I courses cannot get used to the notion of limits.
I have taught such courses and I don't have a reliable method to overcome this barrier.

That is clearly what happened to Mückenheim. Failure to understand the concept of limits is one
of his hang-ups.

If you wanted to help such a student you would reduce the example to its essence, e.g.:

The sequence of numbers 0.1, 0.01, ..., 0.0....1, ... approaches 0 in the limit; limit being defined
in the usual metric of the real line.
In other words, if x_n = 1/10^n then lim x_n = 0 as n -> \infty

Now if the student asks: "Where did the 1 go?", meaning the 1 in the n-th position of the
decimal representation of 10^{-n}, what's the right answer?

Mückenheim thinks the 1 is hiding somewhere in an inaccessible 'dark place'.
And this dotard has spent decades teaching remedial math.

The right answer is: The 1 didn't go anywhere. 1/10^n is still 0.000...1, with a 1 in the
n-th position. And the sequence (1/10^n) is still the sequence (1/10^n) and it's limit
is still and will always be 0.

[Sergi o]

On 11/20/2022 4:23 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 20. November 2022 um 11:11:36 UTC+1:
>> On Sunday, November 20, 2022 at 9:28:15 AM UTC+1, WM wrote:
>>
>> Your BS
>>
>>> suffer[s] from deliberately contradicting basic logic.
>
> In lossless exchange of elements, elements will get lost? Here is the wonderful and simple example:
>
> 1, 2, 3, 4, 5, 6, 7, ...
> 2, 1, 3, 4, 5, 6, 7, ...
> 2, 4, 1, 3, 5, 6, 7, ...
> 2, 4, 6, 1, 3, 5, 7, ...
> and so on.
>
> In every term of the sequence all odd numbers are present and are indexed.

Wrong. you failed to state in above they are in a one to one mapping with the real numbers, and how, very sloppy.

> But "in the limit" they have gone, leaving no traces?

wrong. none of those *infinite* sequences have a limit.


Google mathematical limit

it wont help you at all, good luck

>
> Regards, WM

[Fritz Feldhase]

On Sunday, November 20, 2022 at 3:44:51 PM UTC+1, Sergi o wrote:
> On 11/20/2022 4:23 AM, WM wrote:
> >
> > 1, 2, 3, 4, 5, 6, 7, ...
> > 2, 1, 3, 4, 5, 6, 7, ...
> > 2, 4, 1, 3, 5, 6, 7, ...
> > 2, 4, 6, 1, 3, 5, 7, ...
> > and so on.
> >

> > none of those *infinite* sequences have a limit.

Yes, but the sequence of sequences has one, namely the sequence

2, 4, 6, 8, ...

Now WM is asking where all the odd numbers "have gone", which are in each and every of the sequences mentioned above. Since they aren't *visible* in the limit sequence, they must hide at dark places!

The argument is simple and convincing:

Do you see a pink elephant in the room? No? This *proves* that it is *hiding* at a dark place in the room!

[WM]

JVR schrieb am Sonntag, 20. November 2022 um 14:10:02 UTC+1:
> On Sunday, November 20, 2022 at 11:23:31 AM UTC+1, WM wrote:
> > Fritz Feldhase schrieb am Sonntag, 20. November 2022 um 11:11:36 UTC+1:
> > > On Sunday, November 20, 2022 at 9:28:15 AM UTC+1, WM wrote:
> > >
> > > Your BS
> > >
> > > > suffer[s] from deliberately contradicting basic logic.
> > In lossless exchange of elements, elements will get lost? Here is the wonderful and simple example:
> >
> > 1, 2, 3, 4, 5, 6, 7, ...
> > 2, 1, 3, 4, 5, 6, 7, ...
> > 2, 4, 1, 3, 5, 6, 7, ...
> > 2, 4, 6, 1, 3, 5, 7, ...
> > and so on.
> >
> > In every term of the sequence all odd numbers are present and are indexed.
> > But "in the limit" they have gone, leaving no traces?
> >
> > That would contradict "that every element of the set stands at a definite position of this sequence" and of course basic logic. Note that the number of indices does never change. Obviously there are enough.

> In fact, it happens quite often that students in Analysis I courses cannot get used to the notion of limits.

You have been one of them? Or did you ever teach?

> I have taught such courses and I don't have a reliable method to overcome this barrier.

I doubt that you ever taught (successfully) because then you would know the huge difference between limits in analysis (which are approached but usually not attained) and set theory where all elements have to get indexed - with not a single exception.

>
> If you wanted to help such a student you would reduce the example to its essence, e.g.:
>
> The sequence of numbers 0.1, 0.01, ..., 0.0....1, ... approaches 0 in the limit; limit being defined
> in the usual metric of the real line.
> In other words, if x_n = 1/10^n then lim x_n = 0 as n -> \infty

So it is.

>
> Now if the student asks: "Where did the 1 go?", meaning the 1 in the n-th position of the
> decimal representation of 10^{-n}, what's the right answer?

The 1 did never go. The limit is not attained.

> The right answer is: The 1 didn't go anywhere. 1/10^n is still 0.000...1, with a 1 in the
> n-th position. And the sequence (1/10^n) is still the sequence (1/10^n) and it's limit
> is still and will always be 0.

Right! But the elements of the sequence 1, 2, 3, ... will never diappear. In the above example there is the *limit* without odd nunbers. But this does not mean that odd numbers had gone anywhere. They remain in every term of the sequence, all of them. Alas they cannot be observed when the complete reording has occured. Their positions are dark. This can be see best by my matrices. The initial configuration

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

evolves until

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

has been attained. But never the indexing has been completed. Indexing many fractions together "in the limit" would be undefined and can be excluded. Reducing the discrepancy step by step would imply a first event after finitely many steps. That means that with and without reaching a limit mot fractions have not be indexed as Cantor believed: "then every number p/q comes at an absolutely fixed position of a simple infinite sequence". No that is not possible!

Regards, WM

[WM]

Sergi o schrieb am Sonntag, 20. November 2022 um 15:44:51 UTC+1:
> On 11/20/2022 4:23 AM, WM wrote:

> > 1, 2, 3, 4, 5, 6, 7, ...
> > 2, 1, 3, 4, 5, 6, 7, ...
> > 2, 4, 1, 3, 5, 6, 7, ...
> > 2, 4, 6, 1, 3, 5, 7, ...
> > and so on.
> >
> > In every term of the sequence all odd numbers are present and are indexed.
> Wrong.

Where is the first one missing?

Regards, WM

[WM]

Fritz Feldhase schrieb am Sonntag, 20. November 2022 um 18:20:42 UTC+1:
> On Sunday, November 20, 2022 at 3:44:51 PM UTC+1, Sergi o wrote:
> > On 11/20/2022 4:23 AM, WM wrote:
> > >
> > > 1, 2, 3, 4, 5, 6, 7, ...
> > > 2, 1, 3, 4, 5, 6, 7, ...
> > > 2, 4, 1, 3, 5, 6, 7, ...
> > > 2, 4, 6, 1, 3, 5, 7, ...
> > > and so on.
> > >
> > > none of those *infinite* sequences have a limit.
> Yes, but the sequence of sequences has one, namely the sequence
>
> 2, 4, 6, 8, ...
>
> Now WM is asking where all the odd numbers "have gone", which are in each and every of the sequences mentioned above. Since they aren't *visible* in the limit sequence, they must hide at dark places!

They are and remain visible in every visible term of the sequence.

>
> The argument is simple and convincing:
>
> Do you see a pink elephant in the room? No? This *proves* that it is *hiding* at a dark place in the room!

They cannot leave the set. There were enough places in the beginning, and there remain enough places when all numbers have settled at their final place.

Regards, WM

[WM]

JVR schrieb am Sonntag, 20. November 2022 um 14:10:02 UTC+1:

> The right answer is: The 1 didn't go anywhere.

Very good! It did not go but it can't be localized in any definable term. Now open your eyes and see the darkness.

Regards, WM

[Ben Bacarisse]

JVR <jrenne...@googlemail.com> writes:

> On Sunday, November 20, 2022 at 11:23:31 AM UTC+1, WM wrote:

Note: Dr. Wolfgang Mückenheim or Mueckenheim teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.

>> Fritz Feldhase schrieb am Sonntag, 20. November 2022 um 11:11:36 UTC+1:
>> > On Sunday, November 20, 2022 at 9:28:15 AM UTC+1, WM wrote:
>> >
>> > Your BS
>> >
>> > > suffer[s] from deliberately contradicting basic logic.

>> In lossless exchange of elements, elements will get lost? Here is the
>> wonderful and simple example:
>>
>> 1, 2, 3, 4, 5, 6, 7, ...
>> 2, 1, 3, 4, 5, 6, 7, ...
>> 2, 4, 1, 3, 5, 6, 7, ...
>> 2, 4, 6, 1, 3, 5, 7, ...
>> and so on.
>>
>> In every term of the sequence all odd numbers are present and are indexed.
>> But "in the limit" they have gone, leaving no traces?
>>
>> That would contradict "that every element of the set stands at a
>> definite position of this sequence" and of course basic logic. Note
>> that the number of indices does never change. Obviously there are
>> enough.
>

> In fact, it happens quite often that students in Analysis I courses
> cannot get used to the notion of limits. I have taught such courses
> and I don't have a reliable method to overcome this barrier.

The curious thing is that WM has written a book that includes the
definition of the point-wise limit of the convergent sequence of
functions.

When this last came up, I worked through the definition to show what the
limit function was, and WM flatly rejected it! It's almost as if he
does not understand what's in his book!

--
Ben.

[Ben Bacarisse]

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

> I doubt that you ever taught (successfully) because then you would
> know the huge difference between limits in analysis (which are
> approached but usually not attained) and set theory where all elements
> have to get indexed - with not a single exception.

The sequence of functions, S_k, that I defined converges, point-wise, to
the function E(n) = 2n. Your textbook defines this limit for any
student willing to learn. Is that another error in your book?

You are free to define the final result of the swaps as something other
than this limit, but your refusal to do so is a clear sign that you have
nothing to offer.

--
Ben.

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Samstag, 19. November 2022 um 02:36:14 UTC+1:
>> WM <askas...@gmail.com> writes:

>> > This is the topic:
>> >
>> > Cantor claims that *his* method or process yields a final result:
>> No he does not, and you know it.
>
> You are a liar and I can prove it:

You are dishonestly changing the topic. You said that Cantor does not
even discuss swaps.

>> You have clearly stated that Cantor
>> does not even discuss swaps,
>
> Cantor discusses counting,

But not swaps. The "final result" in question is about the effect of
swaps.

--
Ben.

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 01:03:22 UTC+1:

>> Some detail... There's no need to mess with matrices. For example,
>> consider the trivial sequence
>>
>> S_0(n) = n (i.e. 1, 2, 3, ...)
>>
>> Swapping (1, 2) produces
>>
>> S_1(n) = 2, 1, 3, 4, 5, 6, ....
>>
>> Applying the swap (2, 4) to S_1 produces
>>
>> S_2(n) = 2, 4, 3, 1, 5, 6, ...
>>
>> Each swap of i with 2i yields a sequence with another even number in the
>> initial portion, and no later swap will change the initial segment.
>>
>> So what is the effect of applying the infinite sequence of swaps (i, 2i)
>> to S_0? The natural definition is that it's E(n) = 2n -- the sequence
>> of even numbers. This is simply the limit of a point-wise convergent
>> sequence of functions S_k.
>
> Thank you for this nice example.

You are welcome. You often give very convoluted examples. There are
even simpler examples. The permutations (i,i+1) generate a sequence of
functions that converge to F(n) = n+1. Where did 1 go? It's magic!

> Another proof of dark numbers: The places where the odd numbers end up
> are dark.

Nothing moves anywhere. A swap is just a function and S_k+1 is simply
S_k composed with the k'th permutation function.

Anyway, WMaths does not have dark numbers but the fact that the function
sequence S_k converges, point-wise, to E is a fact of WMaths. There's
no need to add dark numbers to WMaths -- the fact that limit functions
have different image sets to the functions that converge to the limit is
an utterly trivial fact of WMaths (and indeed of proper mathematics)
that needs no magic invention to explain.

--
Ben.

[Sergi o]

one of his serious book reads like a cut and paste job, too many topics, too much depth in short chapters without intro or middle layers, and no one
could know all that math to such depths. (a bad math book, as you need other math books nearby)

[JVR]

Of course, he doesn't understand the material in his book.
Over the years many of the errors in his book have been
discussed in the German math news group and, as you can imagine,
he responds with invective and no insight.

Just take a look at section 33. If provoked, he will even try to defend this
nonsense; sometimes by citing the source from which he copied.

Clearly the guy is endowed with super-human persistence and
very limited understanding.

[JVR]

Any doubts about Mückenheim's intellectual limitations are resolved by
a look at the post at the beginning of this thread.
Apparently, he thinks of that gibberish as a proof of something or other.

[Fritz Feldhase]

On Monday, November 21, 2022 at 8:22:00 AM UTC+1, JVR wrote:

> Any doubts about Mückenheim's intellectual limitations are resolved by
> a look at the post at the beginning of this thread.
> Apparently, he thinks of that gibberish as a proof of something or other.

Well, I'd say it certainly is a "proof" (sort of) of "something or other". :-P

[JVR]

Say a person is faced by two clearly contradictory statements, one being Mückenheim's
proof of something-or-other; the other being the fact that there are 1-1 mappings
from N to NxN.
A sane person would conclude that at least one of the two assertions is incorrect and
would, sooner or later, realize that the latter is demonstrable, hence the former is false.

Obviously, the guy is sick. Maybe not as sick as Atomicus and friends, but still very sick.

[WM]

JVR schrieb am Montag, 21. November 2022 um 09:57:34 UTC+1:

> Say a person is faced by two clearly contradictory statements, one being Mückenheim's
> proof of something-or-other; the other being the fact that there are 1-1 mappings
> from N to NxN.
> A sane person would conclude that at least one of the two assertions is incorrect and
> would, sooner or later, realize that the latter is demonstrable, hence the former is false.
>

A sane person would never have accepted that there is a bijection between all fractions and prime numbers. At least when it is claimed that the bijection depends on arbitrary reordering and can even be inflated to include all algebraic numbers and to exclude nine tenth of all prime numbers. That idea is insane in highest degree!

In addition, it requires to believe that an inclusion monotonic sequence of infinite sets like endsegments has an empty itersection and that exchanging the X's and O's in my example will delete all O's in a definable way but only after all definable terms of the sequence have kept infinitely many O's.

Regards, WM

[WM]

Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:

> The curious thing is that WM has written a book that includes the
> definition of the point-wise limit of the convergent sequence of
> functions.

That is not curious but the only consistent thing to do in potential infinity.

Of course the limit exists. It is the configuration which is approached but never attained. A bijection in actual infinity however is not to be approached only but actually to be realized.

>
> When this last came up, I worked through the definition to show what the
> limit function was, and WM flatly rejected it! It's almost as if he
> does not understand what's in his book!

I do not reject what the limit is! Here is a simple example:

Consider the sequence of strings
21111111111111111111111111...
12111111111111111111111111...
11211111111111111111111111...
11121111111111111111111111...
and so on, exchanging always the 2 and the 1 following next upon the 2.
Will the 2 leave the string? Why should it? There are obviously enough places in actual infinity in the beginning and in every term of the sequence.
Nevertheless, in the limit only the string
11111111111111111111111111...
is visible. Yes that is the limit! The 2, if present yet, occupies a dark place.

Regards, WM

[WM]

Ben Bacarisse schrieb am Montag, 21. November 2022 um 02:13:32 UTC+1:

> WM <askas...@gmail.com> writes:

> > Thank you for this nice example.
> You are welcome. You often give very convoluted examples.

Meanwhile I have even simplified it. See my first post.

> There are
> even simpler examples. The permutations (i,i+1) generate a sequence of
> functions that converge to F(n) = n+1. Where did 1 go? It's magic!

It did not go. But it can't be found after some terms.

> > Another proof of dark numbers: The places where the odd numbers end up
> > are dark.
> Nothing moves anywhere.

The place where it sits changes. That is moving like in a movie.

> A swap is just a function and S_k+1 is simply
> S_k composed with the k'th permutation function.
>
> Anyway, WMaths does not have dark numbers but the fact that the function
> sequence S_k converges, point-wise, to E is a fact of WMaths.

If in the biginning there are not all places, then there is no contradiction if they are not there after all swaps. But if the are all places in the biginning, like in 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, ...
...

then they will remain during all swaps, like the X's and the O's.

>There's
> no need to add dark numbers to WMaths -- the fact that limit functions
> have different image sets to the functions that converge to the limit is
> an utterly trivial fact of WMaths (and indeed of proper mathematics)
> that needs no magic invention to explain.

I cannot but agree heartily. But I am dealing with Cantor's claims about all and every, completely, at an absolutely fixed position, nth number, where not a single one has been forgotten.

If not a single one has been forgotten here
21111111111111111111111111...
then there is no reason to forget it here
11111111111111111111111111...
but it cannot be seen any longer.
Only that is my concern.

Regards, WM

[Sergi o]

each sentence by WM has a lie, red herring, and/or diversion.

[Sergi o]

try writing above in equations in math. Can you do that ?

[Gus Gassmann]

On Sunday, 20 November 2022 at 18:56:36 UTC-4, Ben Bacarisse wrote:
[...]

> When this last came up, I worked through the definition to show what the
> limit function was, and WM flatly rejected it! It's almost as if he
> does not understand what's in his book!

How could he? The material in his book is both original and correct. Problem is, what is correct is copied without understanding, and what is original is incorrect, often grossly so.

[Gus Gassmann]

Bounded above by zero, I'd say.

[WM]

If in every step all O remain which are existing from the beginning and when in the limit they suddenly disappear, then it is impossible to know their indices. Then they do not have indices.

Regards, WM

[Sergi o]

bullpuppy. Every fraction was indexed with the natural numbers at the very start.

So WHAT HAPPENED ?

How did you mess that up ?

In your rabid Swapparoofestavil, you dislodged some of your X pasties and O stickies, they suddenly fell off, and now it is impossible for you to
remember where they go.



> Regards, WM

[WM]

Sergi o schrieb am Montag, 21. November 2022 um 19:32:16 UTC+1:
> On 11/21/2022 12:24 PM, WM wrote:

> > If in every step all O remain which are existing from the beginning and when in the limit they suddenly disappear, then it is impossible to know their indices. Then they do not have indices.
> >
> bullpuppy. Every fraction was indexed with the natural numbers at the very start.

No. This cam be disproved by checking it step by step. In every step all O's remain.
>
> So WHAT HAPPENED ?

All definable indices are applied. The rest is dark.

Regards, WM

[Sergi o]

On 11/21/2022 12:38 PM, WM wrote:
> Sergi o schrieb am Montag, 21. November 2022 um 19:32:16 UTC+1:
>> On 11/21/2022 12:24 PM, WM wrote:
>
>>> If in every step all O remain which are existing from the beginning and when in the limit they suddenly disappear, then it is impossible to know their indices. Then they do not have indices.
>>>
>> bullpuppy. Every fraction was indexed with the natural numbers at the very start.
>
> No. This cam be disproved by checking it step by step.

wrong.

as written by your own hand;

"1. Outline of the proof

(1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions."


so now you disagree and argue with yourself.

>>
>> So WHAT HAPPENED ?
>
> All definable indices are applied. The rest is dark.

no, with you "definable" means "finite". So you stopped again, and now you are in the dark.


>
> Regards, WM

[Ben Bacarisse]

> Of course, he doesn't understand the material in his book.
> Over the years many of the errors in his book have been
> discussed in the German math news group and, as you can imagine,
> he responds with invective and no insight.

So I have come to understand.

But, somewhat ironically, I am interested in the bits that are correct!

The definition he gives of the limit of a point-wise sequence of
functions looks correct to me, and since it's in a book he put his name
to, he can't really deny that the limit of the S_k is E.

> Just take a look at section 33. If provoked, he will even try to defend this
> nonsense; sometimes by citing the source from which he copied.
>
> Clearly the guy is endowed with super-human persistence and
> very limited understanding.

Indeed. I say he can't deny it, but he did the last time I quoted such
a limit back to him. He simply flat-out denied that the limit existed
at all (in WMaths).

--
Ben.

[Ben Bacarisse]

:-) I found an old painting and a violin in the attic. Turns out I had
a Rembrandt and a Stradivarius. Unfortunately, Antonio Stradivari was a
terrible painter, and Rembrandt didn't know the first thing about
making violins. (An old joke, but I couldn't resist.)

--
Ben.

[Ross A. Finlayson]

What's not funny about it?

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Montag, 21. November 2022 um 02:13:32 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>> > Thank you for this nice example.
>> You are welcome. You often give very convoluted examples.
>
> Meanwhile I have even simplified it. See my first post.
>
>> There are
>> even simpler examples. The permutations (i,i+1) generate a sequence of
>> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
>
> It did not go. But it can't be found after some terms.

I don't think you follow the example.

>> > Another proof of dark numbers: The places where the odd numbers end up
>> > are dark.
>> Nothing moves anywhere.
>
> The place where it sits changes. That is moving like in a movie.

No. Movement occurs over time. It's a metaphor you use to bamboozle
your poor students.

--
Ben.

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
>
>> The curious thing is that WM has written a book that includes the
>> definition of the point-wise limit of the convergent sequence of
>> functions.
>
> That is not curious but the only consistent thing to do in potential
> infinity.
>
> Of course the limit exists.

You denied it in 2019.

> It is the configuration which is
> approached but never attained. A bijection in actual infinity however
> is not to be approached only but actually to be realized.
>>
>> When this last came up, I worked through the definition to show what the
>> limit function was, and WM flatly rejected it! It's almost as if he
>> does not understand what's in his book!
>
> I do not reject what the limit is!

Am am glad you have changed your mind, but what I said is true. The
last time this kind of limit came up (May 2019) you flat-out denied that
such function sequences have limits as defined in your book. Do you
want me to give you the message ID where you say "There is no limit!"
and "There is no proper limit function for [...]. Same with your
functions."?

> Here is a simple example:
>
> Consider the sequence of strings
> 21111111111111111111111111...
> 12111111111111111111111111...
> 11211111111111111111111111...
> 11121111111111111111111111...

WMaths does not say how to find the limit of a sequence of strings. But
sequences are functions of N and your book does say how to find the
limit of the functions f_1, f_2, ...

f_k : N -> {1,2}
f_k(n) = 2 if n = k, 1 otherwise

> and so on, exchanging always the 2 and the 1 following next upon the 2.
> Will the 2 leave the string?

For every k, 2 is in the image of f_k. This is a fact of WMaths (and of
proper mathematics).

> Why should it? There are obviously enough places in actual infinity in
> the beginning and in every term of the sequence.

For all k, 2 is in the image of f_k in potential infinity as well as in
proper mathematics.

> Nevertheless, in the limit only the string
> 11111111111111111111111111...
> is visible.

I don't know what visible means in WMaths, but in WMaths the limit is
the function f(n) = 1. This happens to be true in proper mathematics as
well.

> Yes that is the limit! The 2, if present yet, occupies a dark place.

I thought WMaths has no dark places? But don't worry. You don't need
them any more than proper mathematics does. 2 is in the image of every
function in the sequence but /not/ in the image of the limit. If this
is a great mystery of "set theory", it is a mystery of WMaths too. But
I don't find it mysterious at all.

--
Ben.

[WM]

Sergi o schrieb am Montag, 21. November 2022 um 20:23:32 UTC+1:
> On 11/21/2022 12:38 PM, WM wrote:
> > Sergi o schrieb am Montag, 21. November 2022 um 19:32:16 UTC+1:
> >> Every fraction was indexed with the natural numbers at the very start.
> >

> > No. This can be disproved by checking it step by step.

> wrong.
>
> as written by your own hand;
>
> "1. Outline of the proof
>
> (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions."
>
>
> so now you disagree and argue with yourself.

That is called a proof by contradiction.

> > All definable indices are applied. The rest is dark.
> no, with you "definable" means "finite".

Of course.

> So you stopped again,

No, all definable indices are applied, never an index appears with less O's in the matrix.

Regards, WM

[WM]

Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:30:53 UTC+1:

> WM <askas...@gmail.com> writes:

> >> There are
> >> even simpler examples. The permutations (i,i+1) generate a sequence of
> >> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
> >
> > It did not go. But it can't be found after some terms.
> I don't think you follow the example.

Why not use the simplest possible example:

21111111111111111111111111...
12111111111111111111111111...
11211111111111111111111111...
11121111111111111111111111...

and so on, exchanging always the 2 and the 1 following next upon the 2.

Will the 2 leave the string? Why should it? There are obviously enough places.
Nevertheless, in the final state only the string
11111111111111111111111111...
is visible.

> >> Nothing moves anywhere.
> >
> > The place where it sits changes. That is moving like in a movie.
> No. Movement occurs over time.

Movement occurs over different frames or different indices.

> It's a metaphor you use to bamboozle

It is a word that is used in mathematics, for instance by Cantor (that one whose theory we are discussing here): "wo nu alle positiven ganzen Zahlen durchläuft", "wo  Zahlen unserer natürlichen erweiterten Zahlenreihe von  an durchläuft;", "alsdann durchläuft  alle Zahlen von". That movement is meant. The 2 above moves through all indices, all positions of the sequence. There is a limit, but that is not a term of the sequence. The 2 cannot leave the sequence by swapping. It remains - but not at a visible place.

Regards, WM

[WM]

Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
> >
> >> The curious thing is that WM has written a book that includes the
> >> definition of the point-wise limit of the convergent sequence of
> >> functions.
> >
> > That is not curious but the only consistent thing to do in potential
> > infinity.
> >
> > Of course the limit exists.
> You denied it in 2019.

I denied that the limit is attained.

> > It is the configuration which is
> > approached but never attained. A bijection in actual infinity however
> > is not to be approached only but actually to be realized.
> >>
> >> When this last came up, I worked through the definition to show what the
> >> limit function was, and WM flatly rejected it! It's almost as if he
> >> does not understand what's in his book!
> >
> > I do not reject what the limit is!
> Am am glad you have changed your mind, but what I said is true. The
> last time this kind of limit came up (May 2019) you flat-out denied that
> such function sequences have limits as defined in your book. Do you
> want me to give you the message ID where you say "There is no limit!"
> and "There is no proper limit function for [...]. Same with your
> functions."?

Yes.

> > Here is a simple example:
> >
> > Consider the sequence of strings
> > 21111111111111111111111111...
> > 12111111111111111111111111...
> > 11211111111111111111111111...
> > 11121111111111111111111111...
> WMaths does not say how to find the limit of a sequence of strings. But
> sequences are functions of N and your book does say how to find the
> limit of the functions f_1, f_2, ...
>
> f_k : N -> {1,2}
> f_k(n) = 2 if n = k, 1 otherwise
> > and so on, exchanging always the 2 and the 1 following next upon the 2.
> > Will the 2 leave the string?
> For every k, 2 is in the image of f_k. This is a fact of WMaths (and of
> proper mathematics).
> > Why should it? There are obviously enough places in actual infinity in
> > the beginning and in every term of the sequence.
> For all k, 2 is in the image of f_k in potential infinity as well as in
> proper mathematics.

And there are no other terms of the sequence. Nevertheless the 2 cannot be found.

> > Nevertheless, in the limit only the string
> > 11111111111111111111111111...
> > is visible.
> I don't know what visible means in WMaths, but in WMaths the limit is
> the function f(n) = 1. This happens to be true in proper mathematics as
> well.

But the limit is not attained. Nevertheless the 2 cannot be found.
This example is simpler. The infinitely many remaining O in the OP are clearer. They do not leave in any term of the sequence. If they leave in the limit, then there is no indexing possible. If they do not leave, then there is no indexing completed.

> > Yes that is the limit! The 2, if present yet, occupies a dark place.
> I thought WMaths has no dark places? But don't worry. You don't need
> them any more than proper mathematics does. 2 is in the image of every
> function in the sequence but /not/ in the image of the limit.

Same with the infinitely many O's? If they leave in the limit, then there is no indexing possible. If they do not leave, then there is no indexing completed.

> If this
> is a great mystery of "set theory", it is a mystery of WMaths too. But
> I don't find it mysterious at all.

You have never pondered about this fact: If they leave in the limit, then there is no indexing possible. If they do not leave, then there is no indexing completed.

Only this point should be discussed here- and if possible explained in the framework of Cantor's set theory.

Regards, WM

[Fritz Feldhase]

On Tuesday, November 22, 2022 at 12:32:14 PM UTC+1, WM wrote:
> >
> > so now you disagree and argue with yourself.
> >
> That is called

psychosis.

See: "Psychosis is when people lose some contact with reality. This might involve seeing or hearing things that other people cannot see or hear and believing things that are not actually true (delusions)."

https://www.nhs.uk/mental-health/conditions/psychosis/overview/

[Fritz Feldhase]

On Tuesday, November 22, 2022 at 12:51:30 PM UTC+1, WM wrote:

> the 2 cannot be found.

Well, that's to be expected, since it is not "there", you psychotic asshole full of shit.

It would be rather strange if something "could be found" which is not there.

> > If this is a great mystery of "set theory", it is a mystery of WMaths too. But
> > I don't find it mysterious at all.

Same, same.

> You have never pondered about <bla>

Yeah, Mückenheim, whatever.

[Sergi o]

On 11/22/2022 5:41 AM, WM wrote:
> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:30:53 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>>>> There are
>>>> even simpler examples. The permutations (i,i+1) generate a sequence of
>>>> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
>>>
>>> It did not go. But it can't be found after some terms.
>> I don't think you follow the example.
>
> Why not use the simplest possible example:
>
> 21111111111111111111111111...
> 12111111111111111111111111...
> 11211111111111111111111111...
> 11121111111111111111111111...
> and so on, exchanging always the 2 and the 1 following next upon the 2.
> Will the 2 leave the string? Why should it? There are obviously enough places.
> Nevertheless, in the final state

wrong, there is no final state in an infinite sequence.

only the string
> 11111111111111111111111111...
> is visible.

as written 11111111111111111111111111... has no 2 in it.
(google simple math nomenclature for dummies)

>
>>>> Nothing moves anywhere.
>>>
>>> The place where it sits changes. That is moving like in a movie.
>> No. Movement occurs over time.
>
> Movement occurs over different frames or different indices.

there is no movement. stop being lazy and write the entire sequence out.

>
>> It's a metaphor you use to bamboozle
>

> Regards, WM

[WM]

Sergi o schrieb am Dienstag, 22. November 2022 um 15:53:35 UTC+1:
> On 11/22/2022 5:41 AM, WM wrote:
> > Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:30:53 UTC+1:
> >> WM <askas...@gmail.com> writes:
> >
> >>>> There are
> >>>> even simpler examples. The permutations (i,i+1) generate a sequence of
> >>>> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
> >>>
> >>> It did not go. But it can't be found after some terms.
> >> I don't think you follow the example.
> >
> > Why not use the simplest possible example:
> >
> > 21111111111111111111111111...
> > 12111111111111111111111111...
> > 11211111111111111111111111...
> > 11121111111111111111111111...
> > and so on, exchanging always the 2 and the 1 following next upon the 2.
> > Will the 2 leave the string? Why should it? There are obviously enough places.
> > Nevertheless, in the final state
> wrong, there is no final state in an infinite sequence.
> only the string

That is right, there are ℵo dark states following as ... which cannot be distinguished. That is what I call the final state:

> > 11111111111111111111111111...

> > Movement occurs over different frames or different indices.
> there is no movement.

Wrong. Try to understand the frames of a movie.

Regards, WM

[WM]

Fritz Feldhase schrieb am Dienstag, 22. November 2022 um 13:30:49 UTC+1:
> On Tuesday, November 22, 2022 at 12:51:30 PM UTC+1, WM wrote:
>
> > the 2 cannot be found.
> Well, that's to be expected, since it is not "there"
>

> It would be rather strange if something "could be found" which is not there.

It is in ℵo dark places 1111111111... . We cannot know in which.

> > > If this is a great mystery of "set theory", it is a mystery of WMaths too. But
> > > I don't find it mysterious at all.
> Same, same.
>
> > You have never pondered about <bla>
>

That immunizes against thinking and against antinomies. Nevertheless: If the infinitely many O's leave "in the limit", then there is no indexing possible. If they do not leave, then there is no indexing completed.

Regards, WM

[Sergi o]

On 11/22/2022 1:56 PM, WM wrote:
> Sergi o schrieb am Dienstag, 22. November 2022 um 15:53:35 UTC+1:
>> On 11/22/2022 5:41 AM, WM wrote:
>>> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:30:53 UTC+1:
>>>> WM <askas...@gmail.com> writes:
>>>
>>>>>> There are
>>>>>> even simpler examples. The permutations (i,i+1) generate a sequence of
>>>>>> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
>>>>>
>>>>> It did not go. But it can't be found after some terms.
>>>> I don't think you follow the example.
>>>
>>> Why not use the simplest possible example:
>>>
>>> 21111111111111111111111111...
>>> 12111111111111111111111111...
>>> 11211111111111111111111111...
>>> 11121111111111111111111111...
>>> and so on, exchanging always the 2 and the 1 following next upon the 2.
>>> Will the 2 leave the string? Why should it? There are obviously enough places.
>>> Nevertheless, in the final state
>> wrong, there is no final state in an infinite sequence.
>> only the string
>
> That is right, there are ℵo dark states following as ... which cannot be distinguished.

and you are wrong. ... means the sequence continues on indefinitely with the last number repeated.

I'm not going to explain simple Math Nomenclature to you.

no wonder you post so much bullshit around here, you cannot understand the simple Math nomenclature used!


Final Grade: -2 Tard

>
> Regards, WM

[Sergi o]

On 11/22/2022 1:59 PM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 22. November 2022 um 13:30:49 UTC+1:
>> On Tuesday, November 22, 2022 at 12:51:30 PM UTC+1, WM wrote:
>>
>>> the 2 cannot be found.
>> Well, that's to be expected, since it is not "there"
>>
>> It would be rather strange if something "could be found" which is not there.
>
> It is in ℵo dark places 1111111111... . We cannot know in which.

is this it ? ........111111111111111111111111111111112111111111111111111111111111111111111111111111111111..........

see ? you just have to look for it.

Silly.

>
>>>> If this is a great mystery of "set theory", it is a mystery of WMaths too. But
>>>> I don't find it mysterious at all.
>> Same, same.
>>
>>> You have never pondered about <bla>
>>
> That immunizes against thinking

random thoughts-- smells mathy to one, but is fake math.


>
> Regards, WM
>

[Sergi o]

On 11/18/2022 2:37 AM, WM wrote:

> Ben Bacarisse schrieb am Freitag, 18. November 2022 um 02:41:21 UTC+1:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>

>>> I use Cantor's result,
>> No you can't because, as you admit, Cantor does not even talk about
>> swaps, much less define what the state of affairs would be "when all
>> exchanges have been carried out".
>

> Cantor claims that *his* method or process yields a final result: Every natural number is an index of fraction and every fraction has a natural index. In modern terminology: Bijection.

Wrong. Cantor Proved it. you only use Cantors name to lend importance to your tripe.

>>
>> Basically you present a result, based on the currently unspecified
>> notion of "when all exchanges have been carried out", with no
>> justification because you can't say what that magic phrase means.
>
> This same final result would be accomplished by my sequence of matrices, because it is only another language for the same facts.

Wrong. Yours is a finite process applied to infinite sets. Fail.

>

>
> Regards, WM

[WM]

Sergi o schrieb am Mittwoch, 23. November 2022 um 15:40:25 UTC+1:
> On 11/18/2022 2:37 AM, WM wrote:

> > This same final result would be accomplished by my sequence of matrices, because it is only another language for the same facts.
>

> Yours is a finite process applied to infinite sets.

This result holds for *infinitely* many exchanges: Never any X is created, never any O is deleted.

Regards, WM

[Sergi o]

Wrong, You created all the X stickies, and ALL the O pasties.

when you started you put an O stickie on each fraction.

Then you removed it, and put an X pastie on it.

Deny it.


>
> Regards, WM

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:30:53 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>> >> There are
>> >> even simpler examples. The permutations (i,i+1) generate a sequence of
>> >> functions that converge to F(n) = n+1. Where did 1 go? It's magic!
>> >
>> > It did not go. But it can't be found after some terms.
>> I don't think you follow the example.
>
> Why not use the simplest possible example:

I thought it was simple enough, but you didn't appear to get it. But
then the obvious question is why didn't you use the simplest example?
Did you think all the waffling about matrices helps hide the trickery?

> 21111111111111111111111111...
> 12111111111111111111111111...
> 11211111111111111111111111...
> 11121111111111111111111111...
> and so on, exchanging always the 2 and the 1 following next upon the 2.
> Will the 2 leave the string? Why should it? There are obviously enough places.
> Nevertheless, in the final state only the string
> 11111111111111111111111111...
> is visible.
>
>> >> Nothing moves anywhere.
>> >
>> > The place where it sits changes. That is moving like in a movie.
>>
>> No. Movement occurs over time.
>
> Movement occurs over different frames or different indices.

Only metaphorically. And since the metaphor's purpose here is simply to
suggest something "mysterious", it's sole purpose is to confuse your
students.

>> It's a metaphor you use to bamboozle
>
> It is a word that is used in mathematics, for instance by Cantor (that
> one whose theory we are discussing here):

I am discussing WMaths here because the "mystery" is there in your
supposedly corrected mathematics. The image set is the same for every
function, {1,2}, but not for the limit function. Of course there is
nothing mysterious about that, so you need the words: the 2 is only ever
swapped so where does it go in the "final result" -- a term you must
studiously not define in order for you students to think there is a
obvious meaning for it.

After all, even in mathematics, there is no movement and no obvious
"final result", just an infinite sequence of infinite sequences (AKA
functions of N). There is no end to this sequence so we must /invent/ a
meaning for the "final result". You don't like that invention, but the
mathematics is exactly the same in WMaths whether you like it or not.

--
Ben.

[JVR]

On Monday, November 21, 2022 at 11:05:52 AM UTC+1, WM wrote:
> JVR schrieb am Montag, 21. November 2022 um 09:57:34 UTC+1:
>
> > Say a person is faced by two clearly contradictory statements, one being Mückenheim's
> > proof of something-or-other; the other being the fact that there are 1-1 mappings
> > from N to NxN.
> > A sane person would conclude that at least one of the two assertions is incorrect and
> > would, sooner or later, realize that the latter is demonstrable, hence the former is false.
> >
> A sane person would never have accepted that there is a bijection between all fractions and prime numbers. At least when it is claimed that the bijection depends on arbitrary reordering and can even be inflated to include all algebraic numbers and to exclude nine tenth of all prime numbers. That idea is insane in highest degree!
>
> In addition, it requires to believe that an inclusion monotonic sequence of infinite sets like endsegments has an empty itersection and that exchanging the X's and O's in my example will delete all O's in a definable way but only after all definable terms of the sequence have kept infinitely many O's.
>
> Regards, WM

Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
are nonsensical.
Therefore, I congratulate you on the discovery of the undefinable dark numbers, the fallacy of the
binary tree and your numerous other confabulations and leave you to your own devices.

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
>> WM <askas...@gmail.com> writes:
>> > Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
>> >
>> >> The curious thing is that WM has written a book that includes the
>> >> definition of the point-wise limit of the convergent sequence of
>> >> functions.
>> >
>> > That is not curious but the only consistent thing to do in potential
>> > infinity.
>> >
>> > Of course the limit exists.
>> You denied it in 2019.
>
> I denied that the limit is attained.

No, you denied that there was such a limit. Really, your apparent
willingness to lie is shocking. Academics should be happy to
acknowledge mistakes. We all make mistakes, but pretending you did not
repudiate the existence of the limit function is simply dishonest. And
if it's just a lapse of memory (and thus an honest mistake) you should
be more cautious in your remarks.

>> > It is the configuration which is
>> > approached but never attained. A bijection in actual infinity however
>> > is not to be approached only but actually to be realized.
>> >>
>> >> When this last came up, I worked through the definition to show what the
>> >> limit function was, and WM flatly rejected it! It's almost as if he
>> >> does not understand what's in his book!
>> >
>> > I do not reject what the limit is!
>> Am am glad you have changed your mind, but what I said is true. The
>> last time this kind of limit came up (May 2019) you flat-out denied that
>> such function sequences have limits as defined in your book. Do you
>> want me to give you the message ID where you say "There is no limit!"
>> and "There is no proper limit function for [...]. Same with your
>> functions."?
>
> Yes.

Message-ID: <783b4bbe-01ef-4113...@googlegroups.com>

In that message I quote your definition of the limit from page 201 of
your book, and then re-iterate the function sequence I am talking about:

| e_n(k) = [k <= n]

to which /you/ then say:

"There is no limit!"

and two sentences later:

"Same with your functions. It is not correct to calculate limits for
your functions using the mathematics of my book."

I don't think your denial could be any clearer. Anyway, we can correct
your earlier statements now. To what function does the function
sequence e_n converge?

Of course the most obvious way in which you deny such limits is when you
said:

"In fact there exist no limits of non-constant sequences of sets."

in Message-ID: <5912d778-131c-47bb...@googlegroups.com>

But functions are just special relations (page 18 of your book) and
relations from A to B are just subsets of AxB (page 15 of your book) so
/any/ convergent sequence of non-constant functions is a counter
example. They are all non-constant sequences of sets.

(Sadly, one of the many mistakes I make is evident here. I mis-typed
the Iverson bracket expression. The earlier example had e_n(k) = [k >=
n]. Both sequences converge, of course, so it's a detail.)

--
Ben.

[WM]

Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 16:22:36 UTC+1:

> After all, even in mathematics, there is no movement and no obvious
> "final result",

It is needed however if its existence is claimed. Without that claim countability and uncountability could not be proved.

> just an infinite sequence of infinite sequences (AKA
> functions of N). There is no end to this sequence so we must /invent/ a
> meaning for the "final result".

Without the final result of a completely enumerated ℵo-infinite set of real numbers the missing of another real number could not be proved, coulde not even be defined.

> You don't like that invention, but the
> mathematics is exactly the same in WMaths whether you like it or not.

I like that invention. But it forbids that all fractions or other countable sets can be indexed. The limit is not attained. That is just what I prove in the matrix-example. Every definable indexing of a fraction does not reduce the not-indexed fractions. If they get indexed, then only "in the limit" but not as Cantor prescribed it: "every number p/q comes at an absolutely fixed position of a simple infinite sequence". If they don't get indexed, then not all get indexed. But that is Cantor's claim, clearly expressed in case of the algebraic numbers: "thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the nth algebraic number where not a single one of this epitome (ω) has been forgotten."

It is only this endresult which I have disproved.

Regards, WM

[WM]

JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:

> Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> are nonsensical.

Could you help me with your tremendous mathematical knowledge how it would be best to formulate it comprehensible for average mathematicians that
(1) all integer fractions are indexed
(2) using these indices for Cantor's procedure will never reduce the not indexed fractions
(3) not all fractions can be indexed completely in definable way?

Regards, WM

[Fritz Feldhase]

On Friday, November 25, 2022 at 10:15:43 AM UTC+1, WM wrote:
> JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:
> >
> > Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> > are nonsensical.
> >

> Could you help me [...] how it would be best to formulate it comprehensible for average mathematicians

Your nonsense CANNOT be formulated "comprehensible for average mathematicians" (since it is nonsense).

Simple as that.

You see: Obviously, there is nothing we can say that will help you understand why your nonsensical ramblings are nonsensical.

[WM]

Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 17:19:34 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
>
> > Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
> >> WM <askas...@gmail.com> writes:
> >> > Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
> >> >
> >> >> The curious thing is that WM has written a book that includes the
> >> >> definition of the point-wise limit of the convergent sequence of
> >> >> functions.
> >> >
> >> > That is not curious but the only consistent thing to do in potential
> >> > infinity.
> >> >
> >> > Of course the limit exists.
> >> You denied it in 2019.
> >
> > I denied that the limit is attained.
> No, you denied that there was such a limit.

I wrote the 4th edition of my book in 2015. There are limits. Of course they exist. They can be claculated also in set theory, see for instance https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf: Set-theoretical limits of sequences of sets. But they are not attained in the definable way expected by Cantor.

> >> > I do not reject what the limit is!
> >> Am am glad you have changed your mind, but what I said is true. The
> >> last time this kind of limit came up (May 2019) you flat-out denied that
> >> such function sequences have limits as defined in your book. Do you
> >> want me to give you the message ID where you say "There is no limit!"
> >> and "There is no proper limit function for [...]. Same with your
> >> functions."?
> >
> > Yes.
> Message-ID: <783b4bbe-01ef-4113...@googlegroups.com>

Not readable. I would need the link. (Example: The link of the present diccussion is
https://groups.google.com/g/sci.logic/c/zfYogfnYMx0)
But it is not necessary because I do not deny your quotes:

> "There is no limit!"
>
> and two sentences later:
>
> "Same with your functions. It is not correct to calculate limits for
> your functions using the mathematics of my book."

That is true. For set theory you need set-theoretical limits of sequences of sets. But they are not attained in the definable way.

> Of course the most obvious way in which you deny such limits is when you
> said:
>
> "In fact there exist no limits of non-constant sequences of sets."

Again, this means *attained* limits in contrast to calculated limits. Example: The sequence

21111111111111111111111111...
12111111111111111111111111...
11211111111111111111111111...
11121111111111111111111111...
has the limit
11111111111111111111111111...
but does not attain it in a definable way.

Regards, WM

[WM]

Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:
> On Friday, November 25, 2022 at 10:15:43 AM UTC+1, WM wrote:
> > JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:
> > >
> > > Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> > > are nonsensical.
> > >
> > Could you help me [...] how it would be best to formulate it comprehensible for average mathematicians
>
> Your nonsense CANNOT be formulated "comprehensible for average mathematicians"

Not even the first points?

(1) all integer fractions are indexed
(2) using these indices for Cantor's procedure will never reduce the not indexed fractions

So Hilbert was right when he said that mathematics is too difficult for mathematicians , or so.

Regards, WM

[Gus Gassmann]

On Friday, 25 November 2022 at 05:37:39 UTC-4, WM wrote:
> Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:
> > On Friday, November 25, 2022 at 10:15:43 AM UTC+1, WM wrote:
> > > JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:
> > > >
> > > > Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> > > > are nonsensical.
> > > >
> > > Could you help me [...] how it would be best to formulate it comprehensible for average mathematicians
> >
> > Your nonsense CANNOT be formulated "comprehensible for average mathematicians"
> Not even the first points?
> (1) all integer fractions are indexed

For every n in IN, for every m in IN there exists an index function I: IN x IN --> IN and there exists a k in IN such that I(k) = n/m.
You know one of those index function: Cantor's enumeration.

> (2) using these indices for Cantor's procedure will never reduce the not indexed fractions

This is your usual blathering. Of course for every k in IN there are infinitely many k' > k. That and $5 will get you a cup of coffee.

> So Hilbert was right when he said that mathematics is [*WAY*] too difficult for ...

... poor little Mucki.

[WM]

Gus Gassmann schrieb am Freitag, 25. November 2022 um 13:03:23 UTC+1:
> On Friday, 25 November 2022 at 05:37:39 UTC-4, WM wrote:
> > Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:

> > Not even the first points?
> > (1) all integer fractions are indexed
> For every n in IN, for every m in IN there exists an index function I: IN x IN --> IN and there exists a k in IN such that I(k) = n/m.

For point (1) we need only f(1/n) = n.

> You know one of those index function: Cantor's enumeration.

I know that it exists for the first definable numbers. But infinitely many O will not be removed.

> > (2) using these indices for Cantor's procedure will never reduce the not indexed fractions

> Of course for every k in IN there are infinitely many k' > k.

Of course this is already proving that no bijections are existing between infinite sets. But as all the facts are on the table, it is so obvious and elementary that it is usually overlooked and its consequences are not realized. Therefore I have made up my example, where not the least progress is made in *all* definable steps to reduce the discrepancy. That should be comprehensible even for stubborn matheologians.

Regards, WM

[Sergi o]

On 11/25/2022 6:22 AM, WM wrote:
> Gus Gassmann schrieb am Freitag, 25. November 2022 um 13:03:23 UTC+1:
>> On Friday, 25 November 2022 at 05:37:39 UTC-4, WM wrote:
>>> Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:
>
>>> Not even the first points?
>>> (1) all integer fractions are indexed
>> For every n in IN, for every m in IN there exists an index function I: IN x IN --> IN and there exists a k in IN such that I(k) = n/m.
>
> For point (1) we need only f(1/n) = n.

so one point, (1), is many points, f(1/n) = n ??

>
>> You know one of those index function: Cantor's enumeration.
>

> I know that it exists for the first definable *FINITE* numbers.

you put external condition on Cantors enumeration. "definable" means "finite" with WM

> But infinitely many O will not be removed.

yes, the dark circus clowns remain dancing in your head, leaving an O on your face.

>
>>> (2) using these indices for Cantor's procedure will never reduce the not indexed fractions
>> Of course for every k in IN there are infinitely many k' > k.
>

<snip crap>

>
> Regards, WM

[Gus Gassmann]

On Friday, 25 November 2022 at 08:03:23 UTC-4, Gus Gassmann wrote:
> On Friday, 25 November 2022 at 05:37:39 UTC-4, WM wrote:
> > Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:
> > > On Friday, November 25, 2022 at 10:15:43 AM UTC+1, WM wrote:
> > > > JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:
> > > > >
> > > > > Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> > > > > are nonsensical.
> > > > >
> > > > Could you help me [...] how it would be best to formulate it comprehensible for average mathematicians
> > >
> > > Your nonsense CANNOT be formulated "comprehensible for average mathematicians"
> > Not even the first points?
> > (1) all integer fractions are indexed
> For every n in IN, for every m in IN there exists an index function I: IN x IN --> IN and there exists a k in IN such that I(k) = n/m.

That was my own quantifier salad, of course. Should read:

There exists an index function I: IN x IN --> IN such that for every (n, m) in IN x IN, there exists a k in IN such that I(k) = n/m.

[Gus Gassmann]

On Friday, 25 November 2022 at 12:34:32 UTC-4, Gus Gassmann wrote:
> On Friday, 25 November 2022 at 08:03:23 UTC-4, Gus Gassmann wrote:
> > On Friday, 25 November 2022 at 05:37:39 UTC-4, WM wrote:
> > > Fritz Feldhase schrieb am Freitag, 25. November 2022 um 10:30:09 UTC+1:
> > > > On Friday, November 25, 2022 at 10:15:43 AM UTC+1, WM wrote:
> > > > > JVR schrieb am Donnerstag, 24. November 2022 um 16:53:56 UTC+1:
> > > > > >
> > > > > > Obviously, there is nothing I can say that will help you understand why your nonsensical polemics
> > > > > > are nonsensical.
> > > > > >
> > > > > Could you help me [...] how it would be best to formulate it comprehensible for average mathematicians
> > > >
> > > > Your nonsense CANNOT be formulated "comprehensible for average mathematicians"
> > > Not even the first points?
> > > (1) all integer fractions are indexed
> > For every n in IN, for every m in IN there exists an index function I: IN x IN --> IN and there exists a k in IN such that I(k) = n/m.

That was my own quantifier salad, of course, plus non-caffeine operation. Should read:

There exists an index function I: IN x IN --> IN such that for every (n, m) in IN x IN, there exists a k in IN such that I(n/m) = k and I(n, m) =/= I(n', m') if (n', m') =/= (n,m).

[olcott]

On 11/16/2022 12:59 PM, WM wrote:
> Abstract: We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
>
>
> 1. Outline of the proof
>
> (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.

We index every positive fraction's numerator by the columns of a
matrix of natural numbers and its denominator by the rows of this
same matrix.

> (2) Then we distribute, according to Cantor's prescription, these indices over the whole matrix. We observe that in every step prescribed by Cantor the set of indices does not increase and the set of not indexed fractions does not decrease.
> (3) Therefore it is impossible to index all fractions in a definable way. Indexing many fractions "in the limit" would be undefined and can be excluded according to section 2 below.
> (4) After having issued all indices only indexed fractions can be seen in the matrix.
> (5) We conclude from the existing but invisible not indexed fractions that they are sitting at invisible positions, inside of the matrix.
> (6) Hence also the first column of the matrix and therefore also  has invisible, co-called dark elements.
> (7) Hence also the initial mapping of natural numbers and integer fractions cannot have been complete. Bijections, i.e., complete mappings, of infinite sets and are impossible.
>
>
> 2. Rejecting the limit idea
>
> When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" to be completed or that they cannot be completed. Such arguing has to be rejected flatly. For this reason some of Cantor's statements are quoted below.
>
> "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed place of a simple infinite sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126]
>
> "The infinite sequence thus defined has the peculiar property to contain all positive rational numbers and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
>
> "thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one from this epitome () has been forgotten." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]
>
> "such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]
>
> The clearness of these expressions is noteworthy: all and every, at an absolutely fixed place, at a definite position, not a single one has been forgotten.
>
> "In fact, according to the above definition of cardinality, the cardinal number M remains unchanged if in place of an element or of each of some elements, or even of each of all elements m of M another thing is substituted." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 283]
>
> This fact will be utilized to replace the pairs of the bijection by matrices or to attach a matrix to every pair of the bijection, respectively.
>
>
> 3. The proof
>
> If all positive fractions m/n are existing, then they all are contained in 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, ...
>     
>
> If all natural numbers k are existing, then they can be used as indices to index the integer fractions m/1 of the first column. Denoting indexed fractions by X and not indexed fractions by O, we obtain the following matrix:
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> XOOO...
>     
>
> Cantor claimed that all natural numbers k are existing and can be applied to index all positive fractions m/n. They are distributed according to
>
> k = (m + n - 1)(m + n - 2)/2 + m.
> The result is a sequence of fractions
>
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, ... .
>
> This sequence is modelled here in the language of matrices. The indices are taken from their initial positions in the first column and are distributed in the given order.
>
> Index 1 remains at the first term 1/1. The next term 1/2 is indexed with 2 which is taken from position 2/1.
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> XOOO...
>     
>
> Then index 3 it taken from 3/1 and attached to 2/1:
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> XOOO...
>     
>
> Then index 4 it taken from 4/1 and attached to 1/3:
>
> XXXO...
> XOOO...
> OOOO...
> OOOO...
> XOOO...
>     
>
> Then index 5 it taken from 5/1 and attached to 2/2:
>
> XXXO...
> XXOO...
> OOOO...
> OOOO...
> OOOO...
>     
>
> And so on. When finally all exchanges of X and O have been carried through and, according to Cantor, all indices have been issued, it turns out that no fraction without index is visible any longer
>
> XXXX...
> XXXX...
> XXXX...
> XXXX...
> XXXX...
>     
>
> but by the technique of lossless exchange of X and O no O can have left the matrix as long as finite natural numbers are issued as indices. Therefore there are not less fractions without index than at the beginning.
>
> We know that all O and as many fractions without index are remaining, but we cannot find any one. Where are they? Fractions that cannot be found we call dark. The O stay at dark positions. This is the only explanation.
>
> This proof shows that every column has dark positions. Therefore also the integer fractions and also the natural numbers contain dark elements. Cantor's enumerating concerns only the visible fractions, not all fractions. This concerns also every other attempt to enumerate the fractions and even the identical mapping. Bijections, i.e., complete mappings, of infinite sets and |N are impossible.
>
>
> 4. Counterarguments
>
> Sometimes we hear the argument that, in spite of the preconditions explicitly quoted in section 2, a set-theoretical or analytical*) limit should be applied. This however would imply that all the O remain present in all defined matrices until "in the limit" these infinitely many O have to leave in an undefinable way; hence infinitely many fractions have to become indexed such that none of them can be checked - contrary to the proper meaning of indexing.
>
> Some set theorists reject it as inadmissible to limit the indices by starting in the first column. But that means only to check that the set of natural numbers has same size as the set of integer fractions.
>
> Others would tolerate that lossless exchange of X and O can suffer from losses. That argument can be excluded by basic logic.
>
> *) Note that an analytical limit like 0 is approached by the sequence (1/n) but never attained. A bijective mapping of sets however is claimed to be completed, according to section 2.
>
> Regards, WM

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

[olcott]

Proof of the existence of dark numbers (bilingual version)
Wolfgang Mueckenheim
University of Applied Sciences Augsburg
Original publication date: 2022-11-19
https://osf.io/tyvnk/download

[olcott]

On 11/16/2022 12:59 PM, WM wrote:
> Abstract: We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
>
>
> 1. Outline of the proof
>
> (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.

> (2) Then we distribute, according to Cantor's prescription, these indices over the whole matrix. We observe that in every step prescribed by Cantor the set of indices does not increase and the set of not indexed fractions does not decrease.
> (3) Therefore it is impossible to index all fractions in a definable way. Indexing many fractions "in the limit" would be undefined and can be excluded according to section 2 below.

Although there is no way to index every positive fraction with a single
integer index there is a bijection between the integer pair indices of a
matrix to every positive fraction: The column of this matrix would
represent the numerator and the row of this matrix would represent the
denominator.

[Sergi o]

On 11/25/2022 3:34 AM, WM wrote:
> Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 17:19:34 UTC+1:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>
>>> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
>>>> WM <askas...@gmail.com> writes:
>>>>> Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
>>>>>
>>>>>> The curious thing is that WM has written a book that includes the
>>>>>> definition of the point-wise limit of the convergent sequence of
>>>>>> functions.
>>>>>
>>>>> That is not curious but the only consistent thing to do in potential
>>>>> infinity.
>>>>>
>>>>> Of course the limit exists.
>>>> You denied it in 2019.
>>>
>>> I denied that the limit is attained.
>> No, you denied that there was such a limit.
>
> I wrote the 4th edition of my book in 2015. There are limits. Of course they exist. They can be claculated also in set theory, see for instance https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf:

which is a Troll book from postings in sci.math

>
>>>>> I do not reject what the limit is!
>>>> Am am glad you have changed your mind, but what I said is true. The
>>>> last time this kind of limit came up (May 2019) you flat-out denied that
>>>> such function sequences have limits as defined in your book. Do you
>>>> want me to give you the message ID where you say "There is no limit!"
>>>> and "There is no proper limit function for [...]. Same with your
>>>> functions."?
>>>
>>> Yes.
>> Message-ID: <783b4bbe-01ef-4113...@googlegroups.com>
>
> Not readable. I would need the link. (Example: The link of the present diccussion is
> https://groups.google.com/g/sci.logic/c/zfYogfnYMx0)
> But it is not necessary because I do not deny your quotes:
>
>> "There is no limit!"
>>
>> and two sentences later:
>>
>> "Same with your functions. It is not correct to calculate limits for
>> your functions using the mathematics of my book."
>

> That is true. For set theory you need set-theoretical limits of sequences of sets. But they are not attained in the definable *FINITE* way.

>
>> Of course the most obvious way in which you deny such limits is when you
>> said:
>>
>> "In fact there exist no limits of non-constant sequences of sets."
>
> Again, this means *attained* limits in contrast to calculated limits. Example: The sequence
>
> 21111111111111111111111111...
> 12111111111111111111111111...
> 11211111111111111111111111...
> 11121111111111111111111111...
> has the limit
> 11111111111111111111111111...

> but does not attain it in a definable *FINITE* way.


your "definable" means "finite", how can a "finite" sequence attain a limit ?








>
> Regards, WM

[olcott]

On 11/25/2022 2:46 PM, Sergi o wrote:
> On 11/25/2022 3:34 AM, WM wrote:
>> Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 17:19:34 UTC+1:
>>> WM <askas...@gmail.com> writes:
>>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>>> Unendlichen" at Hochschule Augsburg.)
>>>
>>>> Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
>>>>> WM <askas...@gmail.com> writes:
>>>>>> Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36
>>>>>> UTC+1:
>>>>>>
>>>>>>> The curious thing is that WM has written a book that includes the
>>>>>>> definition of the point-wise limit of the convergent sequence of
>>>>>>> functions.
>>>>>>
>>>>>> That is not curious but the only consistent thing to do in potential
>>>>>> infinity.
>>>>>>
>>>>>> Of course the limit exists.
>>>>> You denied it in 2019.
>>>>
>>>> I denied that the limit is attained.
>>> No, you denied that there was such a limit.
>>
>> I wrote the 4th edition of my book in 2015. There are limits. Of
>> course they exist.  They can be claculated also in set theory, see for
>> instance
>> https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf:
>
> which is a Troll book from postings in sci.math

Same author as the author of this post:

Proof of the existence of dark numbers (bilingual version)
Wolfgang Mueckenheim
University of Applied Sciences Augsburg
Original publication date: 2022-11-19
https://osf.io/tyvnk/download

[exfalso.quodlibet]

It is self-evident that there is a bijection between pairs of positive integers and numerator/denominator pairs thus there cannot possibly be a bijection between positive integers and numerator/denominator pairs.

[Jeffrey Rubard]

Oh, "ex falso quodlibet", that's a good rule.
("From a contradiction, infer anything you like.")

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 16:22:36 UTC+1:
>
>> After all, even in mathematics, there is no movement and no obvious
>> "final result",
>
> It is needed however if its existence is claimed. Without that claim
> countability and uncountability could not be proved.

No. The proof that no function from N to R is bijective does not rely
on any such thing. It can be done in WMaths as it can in proper
mathematics.

>> just an infinite sequence of infinite sequences (AKA
>> functions of N). There is no end to this sequence so we must /invent/ a
>> meaning for the "final result".
>
> Without the final result of a completely enumerated ℵo-infinite set of
> real numbers the missing of another real number could not be proved,
> coulde not even be defined.

No, what is needed it a limit and a correct definition of R as being
closed under such limits. You can call that limit the "final result" if
you like but it's just a limit as presented in your book.

>> You don't like that invention, but the
>> mathematics is exactly the same in WMaths whether you like it or not.
>
> I like that invention. But it forbids that all fractions or other
> countable sets can be indexed. The limit is not attained. That is just
> what I prove in the matrix-example.

Limits are rarely attained. What you don't like is the perfectly
reasonable definition that the "final result" can only mean the
un-attained limit. You use the phrase but won't define it because as
soon as you do, the "final result" of your swaps will be the same in
WMaths as it is in proper mathematics. You must keep pretending the I
mean something other than the (usually un-attained) limit. I don't.

--
Ben.

[Ben Bacarisse]

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

> Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 17:19:34 UTC+1:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>
>> > Ben Bacarisse schrieb am Dienstag, 22. November 2022 um 03:31:04 UTC+1:
>> >> WM <askas...@gmail.com> writes:
>> >> > Ben Bacarisse schrieb am Sonntag, 20. November 2022 um 23:56:36 UTC+1:
>> >> >
>> >> >> The curious thing is that WM has written a book that includes the
>> >> >> definition of the point-wise limit of the convergent sequence of
>> >> >> functions.
>> >> >
>> >> > That is not curious but the only consistent thing to do in potential
>> >> > infinity.
>> >> >
>> >> > Of course the limit exists.
>> >> You denied it in 2019.
>> >
>> > I denied that the limit is attained.
>> No, you denied that there was such a limit.

>> >> > I do not reject what the limit is!
>> >> Am am glad you have changed your mind, but what I said is true. The
>> >> last time this kind of limit came up (May 2019) you flat-out denied that
>> >> such function sequences have limits as defined in your book. Do you
>> >> want me to give you the message ID where you say "There is no limit!"
>> >> and "There is no proper limit function for [...]. Same with your
>> >> functions."?
>> >
>> > Yes.
>> Message-ID: <783b4bbe-01ef-4113...@googlegroups.com>
>
> Not readable. I would need the link. (Example: The link of the
> present diccussion is
> https://groups.google.com/g/sci.logic/c/zfYogfnYMx0)

I gave the correct citation for a Usenet article. Google may be gone
tomorrow but that ID will remain valid as long as any archive exists. I
am not going to go to extra effort to find a transient, proprietary link
that you find easier. You were an academic. You should know how to
cite source material.

> But it is not necessary because I do not deny your quotes:
>
>> "There is no limit!"
>>
>> and two sentences later:
>>
>> "Same with your functions. It is not correct to calculate limits for
>> your functions using the mathematics of my book."
>
> That is true. For set theory you need set-theoretical limits of
> sequences of sets. But they are not attained in the definable way.

I gave a convergent sequence and you said there is no limit. You said
the method I used to calculate it (from your book) is not correct for
the functions I gave. Both of these statements were incorrect then, and
they remain incorrect now.

So where do you now stand? To what limit does the sequence e_n
converge? Does it converge to the limit given by the definition in your
book or not? An honest answer to simple technical question would make a
change.

You give (with no technical basis and no calculation) such a limit just
below. Are you using the lack of detail to pretend you sequence
converges but mine, from 2019 does not?

>> Of course the most obvious way in which you deny such limits is when you
>> said:
>>
>> "In fact there exist no limits of non-constant sequences of sets."
>
> Again, this means *attained* limits in contrast to calculated limits.

Don't be silly. You are trying to re-write history. It's dishonest.
Limits exist even when they are not attained. In fact, that is the
usual case for most interesting limits, including the explicit example
of a convergent sequence of set from page 202 of your book.

> Example: The sequence
>
> 21111111111111111111111111...
> 12111111111111111111111111...
> 11211111111111111111111111...
> 11121111111111111111111111...
> has the limit
> 11111111111111111111111111...
> but does not attain it in a definable way.

How do you calculate it? I can show you if you don't know. And why did
you reject my calculations in 2019? Do you now accept the limit I gave
is the correct ones? Will you apologise? Will you give a direct answer
to any of these questions?

--
Ben.

[WM]

_ Olcott schrieb am Freitag, 25. November 2022 um 19:35:10 UTC+1:
> On 11/16/2022 12:59 PM, WM wrote:
> > Abstract: We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
> >
> >
> > 1. Outline of the proof
> >
> > (1) We assume that all natural numbers are existing and are indexing all integer fractions in a matrix of all positive fractions.
> > (2) Then we distribute, according to Cantor's prescription, these indices over the whole matrix. We observe that in every step prescribed by Cantor the set of indices does not increase and the set of not indexed fractions does not decrease.
> > (3) Therefore it is impossible to index all fractions in a definable way. Indexing many fractions "in the limit" would be undefined and can be excluded according to section 2 below.
> Although there is no way to index every positive fraction with a single
> integer index there is a bijection between the integer pair indices of a
> matrix to every positive fraction: The column of this matrix would
> represent the numerator and the row of this matrix would represent the
> denominator.

By the proof of dark numbers every definable bijection of infinite sets is excluded. Dark numbers cannot be mapped in a definable way. They cannot be "taken" as individuals. Not even the identical mapping is definable - although n = n is correct for dark numbers too.

Regards, WM

[WM]

exfalso.quodlibet schrieb am Freitag, 25. November 2022 um 23:52:45 UTC+1:

> It is self-evident that there is a bijection between pairs of positive integers and numerator/denominator pairs thus there cannot possibly be a bijection between positive integers and numerator/denominator pairs.

It is self-evident that x = x for every number, be it visible or dark. But bijections between actually infinite sets don't exist because they contain necessarily dark numbers which cannot be "taken" as individuals.

Regards, WM

[WM]

Ben Bacarisse schrieb am Samstag, 26. November 2022 um 03:04:20 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > Ben Bacarisse schrieb am Donnerstag, 24. November 2022 um 16:22:36 UTC+1:
> >
> >> After all, even in mathematics, there is no movement and no obvious
> >> "final result",
> >
> > It is needed however if its existence is claimed. Without that claim
> > countability and uncountability could not be proved.
> No. The proof that no function from N to R is bijective does not rely
> on any such thing. It can be done in WMaths as it can in proper
> mathematics.

It cannot be done in potential infinity because there is always the idea of Hilbert's hotel. Add the diagonal number to the sequence.

> > Without the final result of a completely enumerated ℵo-infinite set of
> > real numbers the missing of another real number could not be proved,

> > could not even be defined.

> No, what is needed it a limit and a correct definition of R as being
> closed under such limits.

Nonsense. But that is not the topic.

> Limits are rarely attained.

In particular they are not attained when enumerating the fractions. Therefore there is no such enumeration.

> What you don't like is the perfectly
> reasonable definition that the "final result" can only mean the
> un-attained limit.

If the limit is not attained, then the enumeration remains incomplete.

> You must keep pretending the I
> mean something other than the (usually un-attained) limit. I don't.

So you agree that not all fractions are enumerated but you are sure that all fractions are enumerated nevertheless? That's not my logic.

Regards, WM

[WM]

Ben Bacarisse schrieb am Samstag, 26. November 2022 um 03:04:27 UTC+1:

> WM <askas...@gmail.com> writes:

> I gave the correct citation for a Usenet article.

> You were an academic.

I remain so, but I do not use that Net.

> Limits exist even when they are not attained.

Of course. But bijections do not exist if there is more material than can be attained.
In potential infinity there is not more, but in actual infinity there is way more. The dark realm between the potentially infinite definable elements and the limit.

Regards, WM

[Sergi o]

On 11/26/2022 2:06 AM, WM wrote:
> _ Olcott schrieb am Freitag, 25. November 2022 um 19:35:10 UTC+1:
>> On 11/16/2022 12:59 PM, WM wrote:

>>>

>> Although there is no way to index every positive fraction with a single
>> integer index there is a bijection between the integer pair indices of a
>> matrix to every positive fraction: The column of this matrix would
>> represent the numerator and the row of this matrix would represent the
>> denominator.
>

> By the Spoof of dark numbers every definable bijection of infinite sets is excluded. Dark numbers... bla bla
>
> Regards, WM

there is no proof of dark numbers you silly QuaCk!

[Sergi o]

On 11/26/2022 2:09 AM, WM wrote:
> exfalso.quodlibet schrieb am Freitag, 25. November 2022 um 23:52:45 UTC+1:
>
>> It is self-evident that there is a bijection between pairs of positive integers and numerator/denominator pairs thus there cannot possibly be a bijection between positive integers and numerator/denominator pairs.
>
> It is self-evident that x = x for every number, be it visible or dark

WRONG. IAW WM, dark numbers have NO identifying features. therefore, there is no separation between dark numbers, which means there is a dark glob,
and no single dark numbers.

> But bijections between actually infinite sets don't

QuAcK!!!



>
> Regards, WM

[Sergi o]

On 11/26/2022 2:25 AM, WM wrote:

> Ben Bacarisse schrieb am Samstag, 26. November 2022 um 03:04:27 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>> I gave the correct citation for a Usenet article.
>> You were an academic.
>
> I remain so, but I do not use that Net.

You are not a serious academic in Math at all. You are here for your fun.

the posted link is your own OP in this thread, which you forgot.

>
>> Limits exist even when they are not attained.
>
> Of course.

<snip crap>

>
> Regards, WM