William schrieb am Samstag, 4. November 2023 um 23:31:49 UTC+1:
> 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.
William schrieb am Samstag, 4. November 2023 um 23:31:49 UTC+1:
> 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.

Cantor's is the most famous. It is trivial to show and easy to check that my matrices represent precisely Cantor's bijection. You cannot find any natural number which is applied by Cantor but not by me, can you? Therefore the "easy-to-check" part of the Cantor pairing is established by my matrices too, alas beyond all "easy-to-check" pairs there are many unpaired fractions remaining.

Regards, WM

FredJeffries schrieb am Freitag, 3. November 2023 um 22:17:03 UTC+1:
> 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.

That would easily be done by 1 because there is only one set of all fractions.
What he did by k = (m + n - 1)(m + n - 2)/2 + m is counting (or kounting) the fractions one by one - a supertask.

Regards, WM

Jim Burns schrieb am Samstag, 4. November 2023 um 22:34:00 UTC+1:

> 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.

Why is this so? If there is a fixed set with no elements missing and no newly appearing, as Cantor and ZF claim, this woukd be impossible. Only the potential infinity makes the variational mappings possible.

Cantor's is the most famous. It is trivial to show and easy to check that my matrices represent precisely Cantor's bijection. You cannot find any natural number which is applied by Cantor but not by me, can you? Therefore the "easy-to-check" part of the Cantor pairing is established by my matrices too, alas beyond all "easy-to-check" pairs there are many unpaired fractions remaining.

Regards, WM

On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
>You cannot find any natural number which is applied by Cantor but not by me, can you?

Nope you and Cantor use the same elements. For each element used by Cantor your billiard game produces A(n) which does not represent the bijection. Big Deal. There are very many things that do not represent the bijection. You claim that a matrix A has important properties, but you billiard game does not produce A.

William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> >You cannot find any natural number which is applied by Cantor but not by me, can you?
> Nope you and Cantor use the same elements

in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.

> For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.

Neither does Cantor's enumeration.

> Big Deal. There are very many things that do not represent the bijection.

Yes, all attempts are condemned to failure. That was to prove.

> You claim that a matrix A has important properties, but you billiard game does not produce A.

I does not miss any of Cantor's pairs. Only that is important.

Regards, WM

On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> > On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> > >You cannot find any natural number which is applied by Cantor but not by me, can you?
> > Nope you and Cantor use the same elements
> in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.
> > For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.
> Neither does Cantor's enumeration. Cantor uses the set |N, no just elements of the set |N.
> > Big Deal. There are very many things that do not represent the bijection.
> Yes, all attempts are condemned to failure. That was to prove.

Nope. *Some* things are shown to fail. That is all you prove. You claim that a matrix A has important properties (E.g. B is a submatrix or A) bur your billiard game does produce A.

On Sunday, November 5, 2023 at 2:30:35 PM UTC+1, William wrote:

Correction:

> On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> >
> > <bla bla> That was to prove.
> >
> Nope. *Some* things are shown to fail. That is all you prove. You claim that a matrix A has important properties (e.g. B is a submatrix or A) but your billiard game does

not

> produce A.

Perval schrieb am Sonntag, 5. November 2023 um 00:57:52 UTC+1:
> 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.

For infinite sets bijections do not prove equinumerosity. [Bernard Bolzano, creator of the mathematical notion Menge (= set)]

Regards, WM

Continuity and Infinity: Lines and Points and Numbers and Infinity

The consideration of the line/point analog/digital continuous/discrete consideration,
and the numbers that modularly make large numbers parts to infinity,
is that either way neither is the other, but, instead, as from numbers are
from the parts or the lines are from the infinity, then they both ways define
both division and addition, parts and moduli.

This is then the fundamental consideration usually that mathematics
provides "a continuum of a line segment its points" and "numbers that
add up to infinity", both, that continuity is always defined, for the
deductive argument that first so results, mutually, the combined considerations
of completions and incompletions together, bounds, limits, and sums.

Fundamentally this is so, where mathematics provides exactly and only this,
the regular, regular, and then the regular in the discrete and the counting,
of the parts in the units, about the density and the curvature.

Then, there's a recollection of sorts of all theory, logical theory and mathematical
theory about continuous domains, it's simply introduced that there are sorts
continuous domains that make properties of discreteness/continuity,
and discontinuity and so on, the usual same regular properties, augmented
the matching of independent continuous domains, resulting identity a continuous domain.

This involves a usual what would be singular approach to countability and
uncountability, diagonalization and antidiagonalization, then as for the
quantifier disambiguation, for fundamentally quantifier inambiguity,
the common results after comprehension for regular, regular, and regular
theories, and ordinary and extra-ordinary, super-ordinary, as including
all quantifier comprehension, and its existence in wholes and parts from above.

Thusly the only and most usual "continuity and infinity define each other",
or "points and infinity define each other", represents a fair definition,
excluding all the usual arguments that "infinity or infinite-divisibility
don't exist", that they do, the representation of discrete and continuous domains.

William schrieb am Sonntag, 5. November 2023 um 14:30:35 UTC+1:
> On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> > William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> > > On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> > > >You cannot find any natural number which is applied by Cantor but not by me, can you?
> > > Nope you and Cantor use the same elements
> > in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.
> > > For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.
> > Neither does Cantor's enumeration. Cantor uses the set |N, no just elements of the set |N.
> > > Big Deal. There are very many things that do not represent the bijection.
> > Yes, all attempts are condemned to failure. That was to prove.
> Nope. *Some* things are shown to fail.

Cantor's attempt is mimiced by my matrices. Never a matrix position inhabited by a fraction is indexed. This proves that Cantor cannot index all matrix positions.

That is all you prove.

Yes, it is enough. Indexing the fractions is as difficult as indexing the positions.

Regards, WM

On Sunday, November 5, 2023 at 9:26:26 AM UTC-8, Ross Finlayson wrote:
> Continuity and Infinity: Lines and Points and Numbers and Infinity
>
> The consideration of the line/point analog/digital continuous/discrete consideration,
> and the numbers that modularly make large numbers parts to infinity,
> is that either way neither is the other, but, instead, as from numbers are
> from the parts or the lines are from the infinity, then they both ways define
> both division and addition, parts and moduli.
>
> This is then the fundamental consideration usually that mathematics
> provides "a continuum of a line segment its points" and "numbers that
> add up to infinity", both, that continuity is always defined, for the
> deductive argument that first so results, mutually, the combined considerations
> of completions and incompletions together, bounds, limits, and sums.
>
> Fundamentally this is so, where mathematics provides exactly and only this,
> the regular, regular, and then the regular in the discrete and the counting,
> of the parts in the units, about the density and the curvature.
>
>
> Then, there's a recollection of sorts of all theory, logical theory and mathematical
> theory about continuous domains, it's simply introduced that there are sorts
> continuous domains that make properties of discreteness/continuity,
> and discontinuity and so on, the usual same regular properties, augmented
> the matching of independent continuous domains, resulting identity a continuous domain.
>
>
> This involves a usual what would be singular approach to countability and
> uncountability, diagonalization and antidiagonalization, then as for the
> quantifier disambiguation, for fundamentally quantifier inambiguity,
> the common results after comprehension for regular, regular, and regular
> theories, and ordinary and extra-ordinary, super-ordinary, as including
> all quantifier comprehension, and its existence in wholes and parts from above.
>
> Thusly the only and most usual "continuity and infinity define each other",
> or "points and infinity define each other", represents a fair definition,
> excluding all the usual arguments that "infinity or infinite-divisibility
> don't exist", that they do, the representation of discrete and continuous domains.

The zero point no quantity is a beginning....

On Sunday, November 5, 2023 at 9:26:56 AM UTC-8, Transfinity wrote:
> William schrieb am Sonntag, 5. November 2023 um 14:30:35 UTC+1:
> > On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> > > William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> > > > On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> > > > >You cannot find any natural number which is applied by Cantor but not by me, can you?
> > > > Nope you and Cantor use the same elements
> > > in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.
> > > > For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.
> > > Neither does Cantor's enumeration. Cantor uses the set |N, no just elements of the set |N.
> > > > Big Deal. There are very many things that do not represent the bijection.
> > > Yes, all attempts are condemned to failure. That was to prove.
> > Nope. *Some* things are shown to fail.
> Cantor's attempt is mimiced by my matrices. Never a matrix position inhabited by a fraction is indexed. This proves that Cantor cannot index all matrix positions.
> That is all you prove.
> Yes, it is enough. Indexing the fractions is as difficult as indexing the positions.
>
> Regards, WM

This seems the same related rates problems problems.

Here, keep it this way, WM, there is only one variable "t" for time.

Also, it's always a continuous variable.

Now you solve yourself whiling away where to give up.

No, set theory having forgotten continuous time, won't help.

Which are your only fundamental results, ..., off which you make futile balks.

It's a Hodges Howl, "Muckenheim's mis-applications of set theory to his related rates",
basically pointing most-all sections of "Transfinity.pdf to sections in "An editor recalls
some papers, ...".

On Sunday, November 5, 2023 at 9:31:48 AM UTC-8, mitchr...@gmail.com wrote:
> On Sunday, November 5, 2023 at 9:26:26 AM UTC-8, Ross Finlayson wrote:
> > Continuity and Infinity: Lines and Points and Numbers and Infinity
> >
> > The consideration of the line/point analog/digital continuous/discrete consideration,
> > and the numbers that modularly make large numbers parts to infinity,
> > is that either way neither is the other, but, instead, as from numbers are
> > from the parts or the lines are from the infinity, then they both ways define
> > both division and addition, parts and moduli.
> >
> > This is then the fundamental consideration usually that mathematics
> > provides "a continuum of a line segment its points" and "numbers that
> > add up to infinity", both, that continuity is always defined, for the
> > deductive argument that first so results, mutually, the combined considerations
> > of completions and incompletions together, bounds, limits, and sums.
> >
> > Fundamentally this is so, where mathematics provides exactly and only this,
> > the regular, regular, and then the regular in the discrete and the counting,
> > of the parts in the units, about the density and the curvature.
> >
> >
> > Then, there's a recollection of sorts of all theory, logical theory and mathematical
> > theory about continuous domains, it's simply introduced that there are sorts
> > continuous domains that make properties of discreteness/continuity,
> > and discontinuity and so on, the usual same regular properties, augmented
> > the matching of independent continuous domains, resulting identity a continuous domain.
> >
> >
> > This involves a usual what would be singular approach to countability and
> > uncountability, diagonalization and antidiagonalization, then as for the
> > quantifier disambiguation, for fundamentally quantifier inambiguity,
> > the common results after comprehension for regular, regular, and regular
> > theories, and ordinary and extra-ordinary, super-ordinary, as including
> > all quantifier comprehension, and its existence in wholes and parts from above.
> >
> > Thusly the only and most usual "continuity and infinity define each other",
> > or "points and infinity define each other", represents a fair definition,
> > excluding all the usual arguments that "infinity or infinite-divisibility
> > don't exist", that they do, the representation of discrete and continuous domains.
> The zero point no quantity is a beginning....

So, then that's "next" in time, what's "next", it's also nexed if you'll note,
"pointing through a point".

It's one of exactly two things: a line segment end, or, what grows and makes a ball.

A "spiral space-filling curve" or about as simple as a line "next", "nexed layer".

Here, it might almost be simpler to write "nexed" instead of "next" some times,
as for example for "these bare differentials are of course nexed", connected,
then really what you might always relate is the connectedness, of continuity,
that which has none.

(It's both of exactly two things.)

I.e. here your comment then follows "... that is the end."

On Sunday, November 5, 2023 at 9:59:55 AM UTC-8, Ross Finlayson wrote:
> On Sunday, November 5, 2023 at 9:31:48 AM UTC-8, mitchr...@gmail.com wrote:
> > On Sunday, November 5, 2023 at 9:26:26 AM UTC-8, Ross Finlayson wrote:
> > > Continuity and Infinity: Lines and Points and Numbers and Infinity
> > >
> > > The consideration of the line/point analog/digital continuous/discrete consideration,
> > > and the numbers that modularly make large numbers parts to infinity,
> > > is that either way neither is the other, but, instead, as from numbers are
> > > from the parts or the lines are from the infinity, then they both ways define
> > > both division and addition, parts and moduli.
> > >
> > > This is then the fundamental consideration usually that mathematics
> > > provides "a continuum of a line segment its points" and "numbers that
> > > add up to infinity", both, that continuity is always defined, for the
> > > deductive argument that first so results, mutually, the combined considerations
> > > of completions and incompletions together, bounds, limits, and sums.
> > >
> > > Fundamentally this is so, where mathematics provides exactly and only this,
> > > the regular, regular, and then the regular in the discrete and the counting,
> > > of the parts in the units, about the density and the curvature.
> > >
> > >
> > > Then, there's a recollection of sorts of all theory, logical theory and mathematical
> > > theory about continuous domains, it's simply introduced that there are sorts
> > > continuous domains that make properties of discreteness/continuity,
> > > and discontinuity and so on, the usual same regular properties, augmented
> > > the matching of independent continuous domains, resulting identity a continuous domain.
> > >
> > >
> > > This involves a usual what would be singular approach to countability and
> > > uncountability, diagonalization and antidiagonalization, then as for the
> > > quantifier disambiguation, for fundamentally quantifier inambiguity,
> > > the common results after comprehension for regular, regular, and regular
> > > theories, and ordinary and extra-ordinary, super-ordinary, as including
> > > all quantifier comprehension, and its existence in wholes and parts from above.
> > >
> > > Thusly the only and most usual "continuity and infinity define each other",
> > > or "points and infinity define each other", represents a fair definition,
> > > excluding all the usual arguments that "infinity or infinite-divisibility
> > > don't exist", that they do, the representation of discrete and continuous domains.
> > The zero point no quantity is a beginning....
> So, then that's "next" in time, what's "next", it's also nexed if you'll note,
> "pointing through a point".
>
> It's one of exactly two things: a line segment end, or, what grows and makes a ball.
>
> A "spiral space-filling curve" or about as simple as a line "next", "nexed layer".
>
> Here, it might almost be simpler to write "nexed" instead of "next" some times,
> as for example for "these bare differentials are of course nexed", connected,
> then really what you might always relate is the connectedness, of continuity,
> that which has none.
>
> (It's both of exactly two things.)
>
>
> I.e. here your comment then follows "... that is the end."

Spin is based on rotation speed through longer and shorter
spiral distance... it has to pass though more or less distance
for it to appear slow or fast...

On Sunday, November 5, 2023 at 1:26:56 PM UTC-4, Transfinity wrote:
> William schrieb am Sonntag, 5. November 2023 um 14:30:35 UTC+1:
> > On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> > > William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> > > > On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> > > > >You cannot find any natural number which is applied by Cantor but not by me, can you?
> > > > Nope you and Cantor use the same elements
> > > in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.
> > > > For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.
> > > Neither does Cantor's enumeration. Cantor uses the set |N, no just elements of the set |N.
> > > > Big Deal. There are very many things that do not represent the bijection.
> > > Yes, all attempts are condemned to failure. That was to prove.
> > Nope. *Some* things are shown to fail.
> Cantor's attempt

is successful, it produces matrix B, a matrix which represents a bijection between |N and |Nx|N. You do not produce matrix B. You say B is a submatrix of A but your billiard game does not produce A.

On Sunday, November 5, 2023 at 10:07:28 AM UTC-8, mitchr...@gmail.com wrote:
> On Sunday, November 5, 2023 at 9:59:55 AM UTC-8, Ross Finlayson wrote:
> > On Sunday, November 5, 2023 at 9:31:48 AM UTC-8, mitchr...@gmail.com wrote:
> > > On Sunday, November 5, 2023 at 9:26:26 AM UTC-8, Ross Finlayson wrote:
> > > > Continuity and Infinity: Lines and Points and Numbers and Infinity
> > > >
> > > > The consideration of the line/point analog/digital continuous/discrete consideration,
> > > > and the numbers that modularly make large numbers parts to infinity,
> > > > is that either way neither is the other, but, instead, as from numbers are
> > > > from the parts or the lines are from the infinity, then they both ways define
> > > > both division and addition, parts and moduli.
> > > >
> > > > This is then the fundamental consideration usually that mathematics
> > > > provides "a continuum of a line segment its points" and "numbers that
> > > > add up to infinity", both, that continuity is always defined, for the
> > > > deductive argument that first so results, mutually, the combined considerations
> > > > of completions and incompletions together, bounds, limits, and sums..
> > > >
> > > > Fundamentally this is so, where mathematics provides exactly and only this,
> > > > the regular, regular, and then the regular in the discrete and the counting,
> > > > of the parts in the units, about the density and the curvature.
> > > >
> > > >
> > > > Then, there's a recollection of sorts of all theory, logical theory and mathematical
> > > > theory about continuous domains, it's simply introduced that there are sorts
> > > > continuous domains that make properties of discreteness/continuity,
> > > > and discontinuity and so on, the usual same regular properties, augmented
> > > > the matching of independent continuous domains, resulting identity a continuous domain.
> > > >
> > > >
> > > > This involves a usual what would be singular approach to countability and
> > > > uncountability, diagonalization and antidiagonalization, then as for the
> > > > quantifier disambiguation, for fundamentally quantifier inambiguity,
> > > > the common results after comprehension for regular, regular, and regular
> > > > theories, and ordinary and extra-ordinary, super-ordinary, as including
> > > > all quantifier comprehension, and its existence in wholes and parts from above.
> > > >
> > > > Thusly the only and most usual "continuity and infinity define each other",
> > > > or "points and infinity define each other", represents a fair definition,
> > > > excluding all the usual arguments that "infinity or infinite-divisibility
> > > > don't exist", that they do, the representation of discrete and continuous domains.
> > > The zero point no quantity is a beginning....
> > So, then that's "next" in time, what's "next", it's also nexed if you'll note,
> > "pointing through a point".
> >
> > It's one of exactly two things: a line segment end, or, what grows and makes a ball.
> >
> > A "spiral space-filling curve" or about as simple as a line "next", "nexed layer".
> >
> > Here, it might almost be simpler to write "nexed" instead of "next" some times,
> > as for example for "these bare differentials are of course nexed", connected,
> > then really what you might always relate is the connectedness, of continuity,
> > that which has none.
> >
> > (It's both of exactly two things.)
> >
> >
> > I.e. here your comment then follows "... that is the end."
> Spin is based on rotation speed through longer and shorter
> spiral distance... it has to pass though more or less distance
> for it to appear slow or fast...

"Geometry is motion", ....

On 11/4/2023 7:57 PM, Perval wrote:
> El sábado, 4 de noviembre de 2023
> a la(s) 17:34:00 UTC-4, Jim Burns escribió:

>> 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.

By stating _how_ they are different,
you are able to make a claim once which is
true infinitely-many times, once for each
finite set and for each infinite set.

By stating _how_ they are different,
you have arrived at infinity,
metaphorically,
_without_ taking infinitely-many steps.

And you can _continue_ at infinity,
metaphorically,
_without_ taking infinitely-many steps,
looking around, learning, at infinity,
by the method of
making further claims which are
_only not-first-false_

An example of a not-first-false claim is
Q in claim-sequence ⟨... P P⇒Q Q ...⟩
(modus ponens)

Q can be false,
but not if P or P⇒Q is false.
Q cannot be first-false.

You see that Q is not-first-false, despite
not knowing what P or Q mean,
not knowing if P or Q are true or false,
not taking infinitely-many steps.

Consider this claim-sequence:
| 1. S is an infinite set.
| 2. (not-first-false claim about S)
| 3. (not-first-false claim about S)
| ...
| 486,308,636. (not-first-false claim about S)
| 486,308,637. (not-first-false claim about S)

If there is no first-false claim,
then there is no false claim.

Suppose you can see that 2 through 486,308,637
are only not-first-false (as with modus ponens).

If it's true that S is an infinite set,
then there is no first-false claim,
and there is no false claim.

If S is an infinite set
then you know the 486,308,637th claim about S
_without_ taking infinitely-many steps.

> Finite and infinite sets are different.

Yes, they're different.

Saying _how_ they're different is
the start of learning about infinitely-many
_without_ taking infinitely-many steps.

You could say:
| A finite set T can be ordered such that
| first‖last exists in T, and,
| for each split F ᣔ⪻ᣔ H of T,
| last‖first exists in F‖H
| | An infinite set S is not finite, and
| it can't be ordered in that way.

You could follow that claim with
not-first-false claims about T and about S

You would know those claims are true
about T (if finite) and about S (if infinite)
_without_ taking infinitely-many steps, but
by the existence of finitely-many claims.

> 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.

A sequence exists of not-first-false claims
between the premise and the conclusion,
from the description of ℕ the natural numbers
to the claim that ℕ×ℕ can be indexed by ℕ
(Cantor's snake)

I've left out as much as I can of
the claims in the middle.
Even so, I get complaints about too-long posts.

Notice that I don't need to write and
you don't need to read those claims
in order for Cantor's snake to exist.

----
For each ⟨i,j⟩ ∈ ℕ₁×ℕ₁
unique kᵢⱼ ∈ ℕ₁ = ℕ\{0} exists:
kᵢⱼ = i+(i+j-1)(i+j-2)/2

We know this because
we know that '+' '*' are closed in ℕ₁ and
one of (i+j-1) or (i+j-2) is divisible by 2

We know that there isn't double-counting
because we know that,
for each k ∈ ℕ₁
unique ⟨iₖ,jₖ⟩ ∈ ℕ₁×ℕ₁ exists:
iₖ+jₖ = kᐞ := ⌈½+(2⋅k+¼)¹ᐟ²⌉
iₖ = k-(kᐞ-1)(kᐞ-2)/2
jₖ = kᐞ-iₖ

⌈x⌉-1 < x ≤ ⌈x⌉ ∈ ℤ

On Sunday, November 5, 2023 at 10:39:08 AM UTC-8, Ross Finlayson wrote:
> On Sunday, November 5, 2023 at 10:07:28 AM UTC-8, mitchr...@gmail..com wrote:
> > On Sunday, November 5, 2023 at 9:59:55 AM UTC-8, Ross Finlayson wrote:
> > > On Sunday, November 5, 2023 at 9:31:48 AM UTC-8, mitchr...@gmail.com wrote:
> > > > On Sunday, November 5, 2023 at 9:26:26 AM UTC-8, Ross Finlayson wrote:
> > > > > Continuity and Infinity: Lines and Points and Numbers and Infinity
> > > > >
> > > > > The consideration of the line/point analog/digital continuous/discrete consideration,
> > > > > and the numbers that modularly make large numbers parts to infinity,
> > > > > is that either way neither is the other, but, instead, as from numbers are
> > > > > from the parts or the lines are from the infinity, then they both ways define
> > > > > both division and addition, parts and moduli.
> > > > >
> > > > > This is then the fundamental consideration usually that mathematics
> > > > > provides "a continuum of a line segment its points" and "numbers that
> > > > > add up to infinity", both, that continuity is always defined, for the
> > > > > deductive argument that first so results, mutually, the combined considerations
> > > > > of completions and incompletions together, bounds, limits, and sums.
> > > > >
> > > > > Fundamentally this is so, where mathematics provides exactly and only this,
> > > > > the regular, regular, and then the regular in the discrete and the counting,
> > > > > of the parts in the units, about the density and the curvature.
> > > > >
> > > > >
> > > > > Then, there's a recollection of sorts of all theory, logical theory and mathematical
> > > > > theory about continuous domains, it's simply introduced that there are sorts
> > > > > continuous domains that make properties of discreteness/continuity,
> > > > > and discontinuity and so on, the usual same regular properties, augmented
> > > > > the matching of independent continuous domains, resulting identity a continuous domain.
> > > > >
> > > > >
> > > > > This involves a usual what would be singular approach to countability and
> > > > > uncountability, diagonalization and antidiagonalization, then as for the
> > > > > quantifier disambiguation, for fundamentally quantifier inambiguity,
> > > > > the common results after comprehension for regular, regular, and regular
> > > > > theories, and ordinary and extra-ordinary, super-ordinary, as including
> > > > > all quantifier comprehension, and its existence in wholes and parts from above.
> > > > >
> > > > > Thusly the only and most usual "continuity and infinity define each other",
> > > > > or "points and infinity define each other", represents a fair definition,
> > > > > excluding all the usual arguments that "infinity or infinite-divisibility
> > > > > don't exist", that they do, the representation of discrete and continuous domains.
> > > > The zero point no quantity is a beginning....
> > > So, then that's "next" in time, what's "next", it's also nexed if you'll note,
> > > "pointing through a point".
> > >
> > > It's one of exactly two things: a line segment end, or, what grows and makes a ball.
> > >
> > > A "spiral space-filling curve" or about as simple as a line "next", "nexed layer".
> > >
> > > Here, it might almost be simpler to write "nexed" instead of "next" some times,
> > > as for example for "these bare differentials are of course nexed", connected,
> > > then really what you might always relate is the connectedness, of continuity,
> > > that which has none.
> > >
> > > (It's both of exactly two things.)
> > >
> > >
> > > I.e. here your comment then follows "... that is the end."
> > Spin is based on rotation speed through longer and shorter
> > spiral distance... it has to pass though more or less distance
> > for it to appear slow or fast...
> "Geometry is motion", ....

Motion is in geometry.
The Earth moves in elliptical orbit.

Perval explained on 11/4/2023 :
> 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.

Your other sock-puppet already wrote that.

Transfinity was thinking very hard :
>
> FredJeffries schrieb am Freitag, 3. November 2023 um 22:17:03 UTC+1:
>> 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.
>
> That would easily be done by 1 because there is only one set of all
> fractions. What he did by k = (m + n - 1)(m + n - 2)/2 + m is counting (or
> kounting) the fractions one by one - a supertask.

No it doesn't/isn't. It shows, by numerical relations, what a
particular sample bijection looks like. There are others, and your game
isn't one of them. Having one that isn't one of them does not
invalidate the ones that are showing a bijection.

On 11/5/2023 12:24 PM, Transfinity wrote:
> Perval schrieb am Sonntag, 5. November 2023
> um 00:57:52 UTC+1:
>> El sábado, 4 de noviembre de 2023
>> a la(s) 17:34:00 UTC-4, Jim Burns escribió:

>>> 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.
>
> For infinite sets
> bijections do not prove equinumerosity.
> [Bernard Bolzano,
> creator of the mathematical notion Menge (= set)]

For each of set X,
there is a unique element of set Y,
iff
there are at least as many in Y as in X
∃F ⊆ X×Y: ∀x ∈ X, ∃!yₓ ∈ Y: ⟨x,yₓ⟩ ∈ F
⟺
|X| ≤ |Y|
[a paraphrase of Cantor, I'd guess]

If
there are at least as many in Y as in X
and
there are at least as many in X as in Y
then
a bijection between X and Y exists.
[Schroeder-Cantor-Bernstein theorem]
|X| ≤ |Y| ∧ |X| ≥ |Y| ⟹ |X| = |Y|

There are at least as many in Y as in X
or
there are at least as many in X as in Y
|X| ≤ |Y| ∨ |X| ≥ |Y|
[equivalent to Choice]

William schrieb am Sonntag, 5. November 2023 um 19:30:30 UTC+1:
> On Sunday, November 5, 2023 at 1:26:56 PM UTC-4, Transfinity wrote:
> > William schrieb am Sonntag, 5. November 2023 um 14:30:35 UTC+1:
> > > On Sunday, November 5, 2023 at 6:24:36 AM UTC-4, Transfinity wrote:
> > > > William schrieb am Sonntag, 5. November 2023 um 10:23:04 UTC+1:
> > > > > On Sunday, November 5, 2023 at 4:00:47 AM UTC-4, Transfinity wrote:
> > > > > >You cannot find any natural number which is applied by Cantor but not by me, can you?
> > > > > Nope you and Cantor use the same elements
> > > > in the same order and the same extension, and by the same formula k = (m + n - 1)(m + n - 2)/2 + m.
> > > > > For each element used by Cantor your billiard game produces A(n) which does not represent the bijection.
> > > > Neither does Cantor's enumeration. Cantor uses the set |N, no just elements of the set |N.
> > > > > Big Deal. There are very many things that do not represent the bijection.
> > > > Yes, all attempts are condemned to failure. That was to prove.
> > > Nope. *Some* things are shown to fail.
> > Cantor's attempt
> is successful, it produces matrix B

in which step?

>, a matrix which represents a bijection between |N and |Nx|N. You do not produce matrix B.

What natnumber in which step am I short of his?

> You say B is a submatrix of A but your billiard game does not produce A.

If Cantor produces B then I produce it too, because there is no difference between his mapping and my application of the same in a different language. But I prove that no fraction ever sits at an indexed place but all remain in the matrix.

Regards, WM

Jim Burns schrieb am Sonntag, 5. November 2023 um 19:43:54 UTC+1:

> By stating _how_ they are different,

Please state how my counting using the matrix language differs from Cantor's counting. If he gets matrix B I get it too. But there remain fractions in the matrix at uncounted places.

Regards, WM

FromTheRafters schrieb am Sonntag, 5. November 2023 um 22:51:51 UTC+1:
> Transfinity was thinking very hard :
> >
> > FredJeffries schrieb am Freitag, 3. November 2023 um 22:17:03 UTC+1:

> >> My mistake -- Cantor enumerated the SET of fractions.
> >
> > That would easily be done by 1 because there is only one set of all
> > fractions. What he did by k = (m + n - 1)(m + n - 2)/2 + m is counting (or
> > kounting) the fractions one by one - a supertask.
> No it doesn't/isn't. It shows, by numerical relations, what a
> particular sample bijection looks like. There are others, and your game
> isn't one of them.

Please find a difference other than the used language. In particular what part of Matrix B differs in the attempts of Cantor's and mine?

Regards, WM

On Monday, November 6, 2023 at 5:15:10 AM UTC-4, Transfinity wrote:

> I prove that no fraction ever sits at an indexed place

Nope. For each n it is easy to see that some fraction (which fraction may depend on n) "sits at an indexed place" of A(n). Your billiard game never produces a matrix in which no fraction "sits at an indexed place" Your billiard game cannot produce B.