On Friday, November 3, 2023 at 12:15:18 PM UTC-3, Wolfgang wrote:
> William schrieb am Donnerstag, 2. November 2023 um 23:26:07 UTC+1:

> > There is no matrix A in you billiard game.
> Could you name any natnumber that is not used as an index?

Nope. Each natnumber is used to form an A(n). So what. B is not a submatrix of any A(n) There is no matrix A in your billiard game.

On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
> FredJeffries schrieb am Donnerstag, 2. November 2023 um 23:44:52 UTC+1:
> > On Thursday, November 2, 2023 at 3:11:48 PM UTC-7, WM wrote:
>
> > > > > My billiard game mimics Cantor's bijection.
> > > >
> > > > No, it doesn't, because you don't have the second principle of generation of whole real numbers.
> > >
> > > I do exactly the same as Cantor.
> > In that case, you can show us where you use he second principle of generation of whole real numbers.
>
> Cantor does not use it in his counting argument.

Thereby demonstrating that you do not understand it. You cannot even see it..

Dogmatic assertion:
>
> When dealing with Cantor's mappings between infinite sets, it is argued usually that these mappings require a "limit" or "second principle of generation of whole real numbers" to be completed or that they cannot be completed. Such arguing has to be rejected flatly.

Proof that the previous paragraph was a mere dogmatic assertion:
>
> For this reason some of Cantor's statements are quoted below.

Here follows more of your fundamentalist scripture exegesis, requiring absolutely NO understanding of the subject matter nor its context, merely an ability to use a search engine and copy and paste -- a parrot can do it:
>
> "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed position 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 the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
>
> Find a natural number that is applied by Cantor but not by me.

Cantor is not performing the same task as you. Pele did not use any muscle that I do not have, but that doesn't entitle to play in the World Cup.

Cantor enumerated the set of natural numbers. You are spouting gibberish in your childish game.

On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
>
> Find a natural number that is applied by Cantor but not by me.

27,875,364,866,678,986,876,234,834,075,876,233,185,321,078,364,200,367,765,123,233,753,245,243,877,082

On Friday, November 3, 2023 at 12:15:18 PM UTC-3, Wolfgang wrote:
> William schrieb am Donnerstag, 2. November 2023 um 23:26:07 UTC+1:

> > There is no matrix A in you billiard game.
> Could you name any natnumber that is not used as an index?

Nope. Each natural number is used at step n to form A(n). B is not a submatrix of any A(n). There is no matrix A is your billiard game

William schrieb am Freitag, 3. November 2023 um 18:26:10 UTC+1:
> On Friday, November 3, 2023 at 12:15:18 PM UTC-3, Wolfgang wrote:
> > William schrieb am Donnerstag, 2. November 2023 um 23:26:07 UTC+1:
>
> > > There is no matrix A in you billiard game.
> > Could you name any natnumber that is not used as an index?
> Nope. Each natural number is used at step n to form A(n).

That means I use all natural numbers as indices which Cantor used. I equal his idexing.

> B is not a submatrix of any A(n). There is no matrix A is your billiard game

Whatever there might be. I equal his indexing but show that fractions remain without index.

Regards, WM

On Friday, November 3, 2023 at 6:51:36 PM UTC+1, Wolfgang wrote:

> I equal his indexing but

Please see a psychiatrist, soon!

On Friday, November 3, 2023 at 2:51:36 PM UTC-3, Wolfgang wrote:
> >There is no matrix A is your billiard game
> Whatever there might be.

Your game of billiards produces an A(n) for every n, nothing more, nothing less. However, you talk about matrix A . E.g. you claim that B is a submatrix of A.

William schrieb am Freitag, 3. November 2023 um 20:38:51 UTC+1:
> On Friday, November 3, 2023 at 2:51:36 PM UTC-3, Wolfgang wrote:
> > >There is no matrix A is your billiard game
> > Whatever there might be.
> Your game of billiards produces an A(n) for every n, nothing more, nothing less.
Same does Cantors formula, nothing more, nothing less.
In fact nothing more can be done. Every n is attached to a matrix position.

> However, you talk about matrix A .

Cantor talks about indexing all fractions. This is wrong, but that he is applying all indices may be believed.

> E.g. you claim that B is a submatrix of A.

I prove that my result issues all indices as he does. But I *prove* that not all matrix places are indexed (because there are fractions sitting which will ever get indexed).

Regards, WM

FredJeffries schrieb am Freitag, 3. November 2023 um 18:09:30 UTC+1:
> On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:

> > Find a natural number that is applied by Cantor but not by me.
> Cantor is not performing the same task as you.

He issues all natural numbers as indices.

> Cantor enumerated the set of natural numbers.

No, he enumerated fractions, but not enough to confirm his claim.

Regards, WM

FredJeffries schrieb am Freitag, 3. November 2023 um 18:12:43 UTC+1:
> On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
> >
> > Find a natural number that is applied by Cantor but not by me.
> 27,875,364,866,678,986,876,234,834,075,876,233,185,321,078,364,200,367,765,123,233,753,245,243,877,082

Wrong.

Regards, WM

On Friday, November 3, 2023 at 2:09:50 PM UTC-7, Ganzhinterseher wrote:
> FredJeffries schrieb am Freitag, 3. November 2023 um 18:09:30 UTC+1:
> > On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
>
> > > Find a natural number that is applied by Cantor but not by me.
> > Cantor is not performing the same task as you.
> He issues all natural numbers as indices.

Wrong. Cantor uses the SET of natural numbers as the domain of his enumeration function.

> > Cantor enumerated the set of natural numbers.
> No, he enumerated fractions, but not enough to confirm his claim.
>

My mistake -- Cantor enumerated the SET of fractions.

On Friday, November 3, 2023 at 2:11:31 PM UTC-7, Ganzhinterseher wrote:
> FredJeffries schrieb am Freitag, 3. November 2023 um 18:12:43 UTC+1:
> > On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
> > >
> > > Find a natural number that is applied by Cantor but not by me.
> > 27,875,364,866,678,986,876,234,834,075,876,233,185,321,078,364,200,367,765,123,233,753,245,243,877,082
>
> Wrong.
>
Fine. Then show us the step in your 'counting' you apply
27,875,364,866,678,986,876,234,834,075,876,233,185,321,078,364,200,367,765,123,233,753,245,243,877,082

After serious thinking Ganzhinterseher wrote :
> William schrieb am Freitag, 3. November 2023 um 20:38:51 UTC+1:
>> On Friday, November 3, 2023 at 2:51:36 PM UTC-3, Wolfgang wrote:
>>>> There is no matrix A is your billiard game
>>> Whatever there might be.
>> Your game of billiards produces an A(n) for every n, nothing more, nothing
>> less.
> Same does Cantors formula, nothing more, nothing less.
> In fact nothing more can be done. Every n is attached to a matrix position.
>
>> However, you talk about matrix A .
>
> Cantor talks about indexing all fractions. This is wrong, but that he is
> applying all indices may be believed.
>
>> E.g. you claim that B is a submatrix of A.
>
> I prove that my result issues all indices as he does. But I *prove* that not
> all matrix places are indexed (because there are fractions sitting which will
> ever get indexed).

Finding some strange method to show a bijection attempt which doesn't
work, does not invalidate one which does work. Even so, I don't think
you game shows anything at all.

It happens that Ganzhinterseher formulated :
> FredJeffries schrieb am Freitag, 3. November 2023 um 18:09:30 UTC+1:
>> On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
>
>>> Find a natural number that is applied by Cantor but not by me.
>> Cantor is not performing the same task as you.
>
> He issues all natural numbers as indices.
>
>> Cantor enumerated the set of natural numbers.
>
> No, he enumerated fractions, but not enough to confirm his claim.

If he enumerated them, then it was enough.

Ganzhinterseher formulated on Friday :
> FredJeffries schrieb am Freitag, 3. November 2023 um 18:12:43 UTC+1:
>> On Friday, November 3, 2023 at 8:19:51 AM UTC-7, Wolfgang wrote:
>>>
>>> Find a natural number that is applied by Cantor but not by me.
>> 27,875,364,866,678,986,876,234,834,075,876,233,185,321,078,364,200,367,765,123,233,753,245,243,877,082
>
> Wrong.

That 47th digit was a typo.

On Friday, November 3, 2023 at 6:04:53 PM UTC-3, Ganzhinterseher wrote:
> William schrieb am Freitag, 3. November 2023 um 20:38:51 UTC+1:
> > On Friday, November 3, 2023 at 2:51:36 PM UTC-3, Wolfgang wrote:
> > > >There is no matrix A is your billiard game
> > > Whatever there might be.
> > Your game of billiards produces an A(n) for every n, nothing more, nothing less.
> Same does Cantors formula, nothing more, nothing less.
> In fact nothing more can be done. Every n is attached to a matrix position.
> > However, you talk about matrix A .
> Cantor talks about indexing all fractions.

Correctly. Your think this is wrong based on the properties (e.g. you claim A has "dark" positions and from this you conclude A(1) has dark positions) of a matrix A that is not produced by your game of billiards.

it is still a strongest gravity... by density...
Explosion does not belong for a singularity.
Even the BB singularity does not explode.
Why would it? it is infinitely dense...

Mitchell Raemsch

On 11/3/2023 5:04 PM, Ganzhinterseher wrote:
> William schrieb am Freitag,
> 3. November 2023 um 20:38:51 UTC+1:

>> However, you talk about matrix A .
>
> Cantor talks about indexing all fractions.

Cantor talks _finitely_ about
indexing all fractions.
Talking finitely is not a super-task.

> This is wrong, but
> that he is applying all indices
> may be believed.

"All indices" is sufficient.

Let ℕ hold each index.
∀⟨1,…,i⟩ ⊆ ℕ

Some subsets S of ℕ hold each index.
{S⊆ℕ| ∀⟨1,…,i⟩ ⊆ S }

⋂ᣔⁱℕ is the intersection of
each-index-holding S ⊆ ℕ
⋂ᣔⁱℕ = ⋂{S⊆ℕ| ∀⟨1,…,i⟩ ⊆ S }

B ⊆ (⋂ᣔⁱℕ)³
B = {⟨i,j,k⟩|k=i+(i+j-1)(i+j-2)/2}

For each ⟨i,j⟩ ∈ (⋂ᣔⁱℕ)²
exists unique k ∈ ⋂ᣔⁱℕ: ⟨i,j,k⟩ ∈ B
k = i+(i+j-1)(i+j-2)/2
i+j = ⌈½+(2⋅k+¼)¹ᐟ²⌉

For each k ∈ ⋂ᣔⁱℕ
exists unique ⟨i,j⟩ ∈ (⋂ᣔⁱℕ)²: ⟨i,j,k⟩ ∈ B
k = i+(i+j-1)(i+j-2)/2
i+j = ⌈½+(2⋅k+¼)¹ᐟ²⌉

Because arithmetic,
(⋂ᣔⁱℕ)² is the same "size" as ⋂ᣔⁱℕ

WM just showed one more method to prove the opposite of what Cantor wanted to prove.

The problem with Cantor's snake (first diagonal) proof is that it implies that, differently from what happens with finite sets, there are right and wrong methods for mapping infinite sets, depending on the result we want to obtain. If we apply that same principle to the rest of mathematics, they would become inconsistent.

We may decide to map natural and rational numbers by listing rational numbers in order of their denominators, for example; as follows:

1 2 3 …
1/1 1/2 1/3 …

By using this method, we would never be able to map any fraction with a numerator different from one because denominators of one will always be equal to their pair and will increase infinitely as the counting terms increase.

In any other field of exact sciences, we would say that, if two different methods drive to different results, at least one of them must be wrong; and the reason to choose one method instead of another is normally not because it is the one that proves our hypothesis. In the case exposed above, there is no reason to prefer one of the methods better than the other, other than the need of proving a specific hypothesis. Both methods work the same for finite sets, thus, an explanation of why both do not work for infinite sets is required.

Cantor assumes that the above is not a problem in this case because we only need one method that makes the mapping possible in order to affirm that a given set has the same size as the set of natural numbers. We also could claim the opposite; that we only need one method that makes the mapping impossible in order to affirm that a given set has a different size as the set of natural numbers. The choice of one assumption instead of the other is arbitrary, unless a logical explanation invalidates the second method for infinite sets, other than we need it that way to prove a specific hypothesis.

on 11/4/2023, Perval supposed :
> WM just showed one more method to prove the opposite of what Cantor wanted to
> prove.

[...]

....and you *believed* him!?

Still, it only takes one correct pairing demonstration to show that
there *is* a bijection. Such a demonstration shows an equinumerosity of
sets and the fact that other attempts fail to 'show' it is
inconsequential.

On 11/4/2023 1:53 PM, FromTheRafters wrote:
> on 11/4/2023, Perval supposed :
>> WM just showed one more method to prove the opposite of what Cantor
>> wanted to prove.
>
> [...]
>
> ...and you *believed* him!?

Wow! Unless, its a sock puppet talking to itself? Humm...

>
> Still, it only takes one correct pairing demonstration to show that
> there *is* a bijection. Such a demonstration shows an equinumerosity of
> sets and the fact that other attempts fail to 'show' it is inconsequential.

On 11/4/2023 4:28 PM, Perval wrote:

> WM just showed one more method to prove
> the opposite of what Cantor wanted to prove.

I think you and I disagree about
what Cantor wanted to prove.

> The problem with Cantor's snake (first diagonal)
> proof is that it implies that,
> differently from what happens with finite sets,
> there are right and wrong methods for mapping
> infinite sets,
> depending on the result we want to obtain.

I think that "right and wrong methods" isn't
a problem. I think it is what Cantor wanted to
prove: Infinite sets and finite sets are
different in that way.

For a finite set,
each one-to-one mapping is "right": meaning,
it maps the finite set to itself exactly.

For an infinite set,
not all one-to-one mappings map
the infinite set to itself exactly.

I don't want to call such a mapping "wrong".
It is what it is.
What would be wrong would be talking about
an _infinite_ set as though it was _finite_
as though each one-to-one mapping mapped
to itself exactly.

The problem with Cantor's snake,
the only problem I see,
is that
our set-intuition is a distillation of
our experience of _finite_ sets, and thus
it leads us astray outside of that experience.

But that's why mathematics classes have exercises.
New experiences to be distilled into new intuitions.

> If we apply that same principle to
> the rest of mathematics,
> they would become inconsistent.

If we apply the description of infinite sets
to finite sets, that would be inconsistent.
If that's what you mean, yes.

Therefore, we shouldn't describe finite sets
incorrectly. Or infinite sets, either.

But that isn't a problem, any more than
taking about a triangle as though it had
four corners is a problem.
We shouldn't do that, but we (most of us) don't.

> In any other field of exact sciences,
> we would say that,
> if two different methods drive to different results,
> at least one of them must be wrong;

When we have reason to think that
different results should be the same,
_something_ is wrong.

For finite sets, we have reason to think that.

For infinite sets, we don't.

In other disciplines,
which possibility of many is wrong
can be a very interesting question.
When the Michelson-Morley experiment
disagreed with Newton, we decided that
it was absolute space and time which was wrong.

Deciding that infinite sets must be wrong
because they aren't like finite sets
strikes me as the same kind of reaction
as deciding that the Michelson-Morely
experiment is "wrong" somehow, to be determined,
because absolute space and time must be right.

> Cantor assumes that
> the above is not a problem in this case
> because we only need one method that makes
> the mapping possible in order to affirm that
> a given set has the same size as
> the set of natural numbers.

Cantor _lets go_ of the assumption that
all sets are like finite sets.
I see parallels to letting go of the assumption
that space and time are absolute.

It is trivial to show there is a bijection from a Peano set to a proper subset.

Yes, there is more than one bijection from the naturals to the rationals.

El sábado, 4 de noviembre de 2023 a la(s) 17:34:00 UTC-4, Jim Burns escribió:
> On 11/4/2023 4:28 PM, Perval wrote:

> For a finite set,
> each one-to-one mapping is "right": meaning,
> it maps the finite set to itself exactly.
>
> For an infinite set,
> not all one-to-one mappings map
> the infinite set to itself exactly.

Finite and infinite sets are different.
Thus, for all infinite sets, results from Cantor's snake are valid and from all other methods are invalid.
Something is missing between the premise and the conclusion.

El sábado, 4 de noviembre de 2023 a la(s) 17:34:00 UTC-4, Jim Burns escribió:
> On 11/4/2023 4:28 PM, Perval wrote:
>
> For a finite set,
> each one-to-one mapping is "right": meaning,
> it maps the finite set to itself exactly.
>
> For an infinite set,
> not all one-to-one mappings map
> the infinite set to itself exactly.

Finite and infinite sets are different.
Thus, for infinite sets, results from Cantor's snake are true and from other methods are false.
Something is missing between the premise and the conclusion.