From Newsgroup: sci.math

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller
number than the infinite number of rational numbers,

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

because it's possible to create a one-to-one mapping of every rational
number to every integer.

Otherwise correct (as purportedly proven by Georg Cantor at the end of the 19th/beginning of the 20th century): |ran| = |raU| = ra|reC.

The misconception that this would not be so can arise from the assumption
that raU = ran |u ran. But actually, raU ree ran |u ran since e.g. 2/2 = 1/1 and
ran/0 ree raU as raU := {p/q : p, q ree ran, q > 0}.

But then |raU| < |ran |u ran|; and while |ran| < |ran |u ran|, |ran| < |raU| does NOT follow,
and is fact false: |ran| = |raU| < |ran |u ran|.

ISTM that Bill Sloman's statement would be true when comparing the cardinalities of ran (or raU) and raY, the set of _real_ numbers, instead. ran (and
raU) is/are countable (countably infinite), while raY is uncountable (uncountably infinite, as also purportedly proven by Cantor). |ran| = |raU| = ra|reC; |raY| = 2^ra|reC, and (ISTM uncontroversial that) ra|reC < 2^ra|reC, so then
|ran| = |raU| < |raY|.

<https://en.wikipedia.org/wiki/Aleph_number>
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/21/2026 07:16 AM, Thomas 'PointedEars' Lahn wrote:

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller
number than the infinite number of rational numbers,

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

because it's possible to create a one-to-one mapping of every rational
number to every integer.

Otherwise correct (as purportedly proven by Georg Cantor at the end of the 19th/beginning of the 20th century): |ran| = |raU| = ra|reC.

The misconception that this would not be so can arise from the assumption that raU = ran |u ran. But actually, raU ree ran |u ran since e.g. 2/2 = 1/1 and
ran/0 ree raU as raU := {p/q : p, q ree ran, q > 0}.

But then |raU| < |ran |u ran|; and while |ran| < |ran |u ran|, |ran| < |raU| does NOT follow,
and is fact false: |ran| = |raU| < |ran |u ran|.

ISTM that Bill Sloman's statement would be true when comparing the cardinalities of ran (or raU) and raY, the set of _real_ numbers, instead. ran (and
raU) is/are countable (countably infinite), while raY is uncountable (uncountably infinite, as also purportedly proven by Cantor). |ran| = |raU| =
ra|reC; |raY| = 2^ra|reC, and (ISTM uncontroversial that) ra|reC < 2^ra|reC, so then
|ran| = |raU| < |raY|.

<https://en.wikipedia.org/wiki/Aleph_number>

Cardinality is rather _less precise_ than other matters
of size relation like, for example, asymptotic density.

Or, "half of the integers are even".

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is rather specific to sets, and, set theory
rather _describes_ numbers than _is_ numbers,
that though "descriptive set theory" is a great account
of formalization in mathematics.

Emil duBois-Reymond discovered various arguments for
the uncountability of reals, later Cantor wrote them
in set theory.




--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/21/2026 07:16 AM, Thomas 'PointedEars' Lahn wrote:

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller
number than the infinite number of rational numbers,

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

because it's possible to create a one-to-one mapping of every rational
number to every integer.

Otherwise correct (as purportedly proven by Georg Cantor at the end of the 19th/beginning of the 20th century): |ran| = |raU| = ra|reC.

The misconception that this would not be so can arise from the assumption that raU = ran |u ran. But actually, raU ree ran |u ran since e.g. 2/2 = 1/1 and
ran/0 ree raU as raU := {p/q : p, q ree ran, q > 0}.

But then |raU| < |ran |u ran|; and while |ran| < |ran |u ran|, |ran| < |raU| does NOT follow,
and is fact false: |ran| = |raU| < |ran |u ran|.

ISTM that Bill Sloman's statement would be true when comparing the cardinalities of ran (or raU) and raY, the set of _real_ numbers, instead. ran (and
raU) is/are countable (countably infinite), while raY is uncountable (uncountably infinite, as also purportedly proven by Cantor). |ran| = |raU| =
ra|reC; |raY| = 2^ra|reC, and (ISTM uncontroversial that) ra|reC < 2^ra|reC, so then
|ran| = |raU| < |raY|.

<https://en.wikipedia.org/wiki/Aleph_number>

Cardinality is rather _less precise_ than other matters
of size relation like, for example, asymptotic density.

Or, "half of the integers are even".

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is rather specific to sets, and, set theory
rather _describes_ numbers than _is_ numbers,
that though "descriptive set theory" is a great account
of formalization in mathematics.

Emil duBois-Reymond discovered various arguments for
the uncountability of reals, later Cantor wrote them
in set theory.


About the Continuum Hypothesis of G. Cantor, there's
that Goedel showed it consistent one way and von Neumann
another, then P. Cohen added an axiom to make it
independent instead of inconsistent, set theory.



--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/21/2026 08:14 AM, Ross Finlayson wrote:

On 02/21/2026 07:16 AM, Thomas 'PointedEars' Lahn wrote:

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller >>>> number than the infinite number of rational numbers,

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

because it's possible to create a one-to-one mapping of every rational
number to every integer.

Otherwise correct (as purportedly proven by Georg Cantor at the end of
the
19th/beginning of the 20th century): |ran| = |raU| = ra|reC.

The misconception that this would not be so can arise from the assumption
that raU = ran |u ran. But actually, raU ree ran |u ran since e.g. 2/2 = 1/1 and
ran/0 ree raU as raU := {p/q : p, q ree ran, q > 0}.

But then |raU| < |ran |u ran|; and while |ran| < |ran |u ran|, |ran| < |raU| does NOT
follow,
and is fact false: |ran| = |raU| < |ran |u ran|.

ISTM that Bill Sloman's statement would be true when comparing the
cardinalities of ran (or raU) and raY, the set of _real_ numbers, instead. >> ran (and
raU) is/are countable (countably infinite), while raY is uncountable
(uncountably infinite, as also purportedly proven by Cantor). |ran| =
|raU| =
ra|reC; |raY| = 2^ra|reC, and (ISTM uncontroversial that) ra|reC < 2^ra|reC, so then
|ran| = |raU| < |raY|.

<https://en.wikipedia.org/wiki/Aleph_number>

Cardinality is rather _less precise_ than other matters
of size relation like, for example, asymptotic density.

Or, "half of the integers are even".

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is rather specific to sets, and, set theory
rather _describes_ numbers than _is_ numbers,
that though "descriptive set theory" is a great account
of formalization in mathematics.

Emil duBois-Reymond discovered various arguments for
the uncountability of reals, later Cantor wrote them
in set theory.

About the Continuum Hypothesis of G. Cantor, there's
that Goedel showed it consistent one way and von Neumann
another, then P. Cohen added an axiom to make it
independent instead of inconsistent, set theory.

As one might imagine, that's a bit messy, since then
thusly one may derive contradictions in set theory
itself, and not even talking about how to derive
contradictions in set theory about description of
other theories of one relation, like ordinals for
order theory or about class/set distinction, or
about theories of other objects like those of
geometry or number theory, as modeled in
ordinary set theory.


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/21/2026 08:30 AM, Ross Finlayson wrote:

On 02/21/2026 08:14 AM, Ross Finlayson wrote:

On 02/21/2026 07:16 AM, Thomas 'PointedEars' Lahn wrote:

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a
smaller
number than the infinite number of rational numbers,

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

because it's possible to create a one-to-one mapping of every rational >>>> number to every integer.

Otherwise correct (as purportedly proven by Georg Cantor at the end of
the
19th/beginning of the 20th century): |ran| = |raU| = ra|reC.

The misconception that this would not be so can arise from the
assumption
that raU = ran |u ran. But actually, raU ree ran |u ran since e.g. 2/2 = 1/1 and
ran/0 ree raU as raU := {p/q : p, q ree ran, q > 0}.

But then |raU| < |ran |u ran|; and while |ran| < |ran |u ran|, |ran| < |raU| does NOT
follow,
and is fact false: |ran| = |raU| < |ran |u ran|.

ISTM that Bill Sloman's statement would be true when comparing the
cardinalities of ran (or raU) and raY, the set of _real_ numbers, instead. >>> ran (and
raU) is/are countable (countably infinite), while raY is uncountable
(uncountably infinite, as also purportedly proven by Cantor). |ran| =
|raU| =
ra|reC; |raY| = 2^ra|reC, and (ISTM uncontroversial that) ra|reC < 2^ra|reC, so then
|ran| = |raU| < |raY|.

<https://en.wikipedia.org/wiki/Aleph_number>

Cardinality is rather _less precise_ than other matters
of size relation like, for example, asymptotic density.

Or, "half of the integers are even".

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is rather specific to sets, and, set theory
rather _describes_ numbers than _is_ numbers,
that though "descriptive set theory" is a great account
of formalization in mathematics.

Emil duBois-Reymond discovered various arguments for
the uncountability of reals, later Cantor wrote them
in set theory.


About the Continuum Hypothesis of G. Cantor, there's
that Goedel showed it consistent one way and von Neumann
another, then P. Cohen added an axiom to make it
independent instead of inconsistent, set theory.

As one might imagine, that's a bit messy, since then
thusly one may derive contradictions in set theory
itself, and not even talking about how to derive
contradictions in set theory about description of
other theories of one relation, like ordinals for
order theory or about class/set distinction, or
about theories of other objects like those of
geometry or number theory, as modeled in
ordinary set theory.

Or, I suppose that was Paul if not Emil
duBois-Reymond, mea culpa.


It's also for duBois-Reymond the idea of
all the expressions of real-valued variables,
in the language of those then and by their
differences in the asymptotic, then that
these each cross the line at zero the abscissa,
the "long line" of duBois-Reymond, is only everywhere
crossing the line itself, so is a continuous domain,
while though its cardinally larger than the usual
definition of the Archimedean complete ordered field,
usually written "R" in blackboard-bold font.

Then, that the "line-reals" or "drawing the line",
"line-drawing", is the usual account of that
drawing a line makes a line segment each as of
points, in a line: these "iota-values" are
also a continuous domain, with extent density
completeness measure, though, that's countable,
not uncountable.

How then that's not inconsistent according to
set theory's models of these as different sets
that have the same topological properties,
is simply enough for line-reals the function
establishing them, a "natural/unit equivalency
function" for their cardinal equivalency or equipollency,
is simply enough not a Cartesian function, then
that besides itself falling out of the results otherwise
for un-countability as not disqualified and rejected,
then as non-Cartesian isn't connected transitively,
to be disqualified and rejected as a bijection between
ordinary naturals and a bounded continuous domain.


This isn't usually brought up in class, yet,
it's an exercise you can verify yourself.
For example, I regularly have put it to
large, competent, conscientious, co-operative reasoners.




--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 21.02.2026 um 16:16 schrieb Thomas 'PointedEars' Lahn:

[X-Post & F'up2 sci.math]

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller
number than the infinite number of rational numbers,

Of course. There are reals which are not integers but every integer is a
real.

"One of the great challenges in this world is knowing enough about
a subject to think you're right, but not enough about the subject
to know you're wrong."

That's the case with all conbvinced set theorists. They claim actual
infinity (or deny to think about that at all) but don't know that their bijections concern only potential infinity. In every case they claim
that their sets are complete.

--Neil deGrasse Tyson, astrophysicist and science communicator
(in his MasterClass promotion video:
<https://www.youtube.com/watch?v=io6QdGcoWMU>)

(SCNR)

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

Even more precisely, ra|reC is a useless notion.

because it's possible to create a one-to-one mapping of every rational
number to every integer.

That is impossible.

Proof: According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence

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

Its terms can be represented by matrices. When we attach all indeXes k =
1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1
and indicate missing indexes by hOles O, then we get the matrix M(0) as starting position:

XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
... ... ... ...
M(0) M(2) M(3) M(4) ...

M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index 2
from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3. Successively all fractions of the sequence get indexed. In the limit,
denoted by M(reR), we see no fraction without index remaining. Note that
the only difference to Cantor's enumeration is that Cantor does not
render account for the source of the indices.

Every X, representing the index k, when taken from its present fraction
m/n, is replaced by the O taken from the fraction to be indexed by this
k. Its last carrier m/n will be indexed later by another index.
Important is that, when continuing, no O can leave the matrix as long as
any index X blocks the only possible drain, i.e., the first column. And
if leaving, where should it settle?

As long as indexes are in the drain, no O has left. The presence of all
O indicates that almost all fractions are not indexed. And after all
indexes have been issued and the drain has become free, no indexes are available which could index the remaining matrix elements, yet covered by O.

It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized. Lossless transpositions cannot suffer losses. The limit matrix M(reR) only shows what should have happened when
all fractions were indexed. Logic proves that this cannot have happened
by exchanges. The only explanation for finally seeing M(reR) is that there
are invisible matrix positions, existing already at the start. Obviously
by exchanging O and X no O can leave the matrix, but the O can disappear
by moving without end, from visible to invisible positions.

The number of not indexed fractions remains |rao|*(|rao|-1) for all
definable terms of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3,
3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
Hence |rao|*(|rao|-1) fractions cannot be indexed by definable indices.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

Example: In counting the rationals like in Hilbert's hotel always
another guest can be additionally accomodated. But in Cantor's diagonal
proof it is excluded that the antidiagonal number can be additionally accomodated.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense. They
looked like promising concepts during the early development of set
theory, but as the decades went on, they faded into the meaninglessness
they occupy today.

[ .... ]

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.
Potential infinity: Hilbert's hotel. Every guest can be accomodated.
Actual infinity: Cantor's diagonal argument. The antidiagonal cannot be accomodated.

They
looked like promising concepts during the early development of set
theory, but as the decades went on, they faded into the meaninglessness
they occupy today.

They are not meaningless but their clear recognition shows that set
theory is inconsistent. Therefore they are banned.

Regards, WM


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

wm wrote:

Am 21.02.2026 um 16:16 schrieb Thomas 'PointedEars' Lahn:

Jeroen Belleman wrote:

On 2/20/26 19:31, Bill Sloman wrote:

I know enough to know that the infinite number of integers is a smaller >>>> number than the infinite number of rational numbers,

Of course.

There is nothing "of course" about this. That claim is simply false, a (common) misconception, *as I had just proven*.

There are reals which are not integers but every integer is a real.

Bill Sloman was not talking about the set of *real* numbers. Learn to read.

but I don't get excited about it.

I don't think that is correct. Both the sets of natural and rational
numbers are aleph-0 in size,

More precisely, their _cardinality_ is ra|reC (strictly: _alef_-0).

Even more precisely, ra|reC is a useless notion.

No, it is not. You do not understand it, but that does not make it useless.

because it's possible to create a one-to-one mapping of every rational
number to every integer.

That is impossible.

No, it is not.

Proof: According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

So far that is only *your* claim, not Cantor's. You have not provided any evidence to support your claim. And, in fact, after you insisted that
Cantor had used it in a letter that he wrote to Lipschitz in 1883, I have obtained the text of that letter, and showed (you) that in that very letter Cantor suggests a completely different method to enumerate and thereby count the rational numbers than you claimed he would. [1]

Yet you keep denying that.

[1] <mid:10mj6iv$9pdp$1@gwaiyur.mb-net.net>
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

Potential infinity: Hilbert's hotel. Every guest can be accomodated.
Actual infinity: Cantor's diagonal argument. The antidiagonal cannot be accomodated.

Typical crank vagueness and meaninglessness.

They looked like promising concepts during the early development of
set theory, but as the decades went on, they faded into the
meaninglessness they occupy today.

They are not meaningless but their clear recognition shows that set
theory is inconsistent. Therefore they are banned.

If they are inconsistent with set theory, that would be a good reason to
banish them. They are in any case not useful. I doubt they are even rigorously defined.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/22/2026 02:08 PM, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

Potential infinity: Hilbert's hotel. Every guest can be accomodated.
Actual infinity: Cantor's diagonal argument. The antidiagonal cannot be
accomodated.

Typical crank vagueness and meaninglessness.

They looked like promising concepts during the early development of
set theory, but as the decades went on, they faded into the
meaninglessness they occupy today.

They are not meaningless but their clear recognition shows that set
theory is inconsistent. Therefore they are banned.

If they are inconsistent with set theory, that would be a good reason to banish them. They are in any case not useful. I doubt they are even rigorously defined.

Regards, WM

I'll agree that WM is a rather useless retro-finitist,
and I think it's just another sock-puppet bot of the
burse-scheiss variety.

That said though, my greater account of the relevance
of cardinality to infinity and size relations of numbers
in sets is quite thoroughly involved and correct.

And super-standard.


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

Proof: According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

So far that is only *your* claim, not Cantor's. You have not provided any evidence to support your claim.

I told you once and I told you twice
But you never listen to my advice
Well, this is the last time:
[Cantor, Collected Works, p. 132]

And, in fact, after you insisted that
Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals
where the multiple appearances had been deleted by hand.

I have
obtained the text of that letter, and showed (you) that in that very letter Cantor suggests a completely different method to enumerate and thereby count the rational numbers

He had used his formula and then deleted the multiple appearances by
hand - a supertask like enumerating the algebraics.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 22.02.2026 um 23:08 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

I understand that these deeper notions are prohibited by the leading
swindlers because they know that this knowledge disproves set theory.

Potential infinity: Hilbert's hotel. Every guest can be accomodated.
Actual infinity: Cantor's diagonal argument. The antidiagonal cannot be
accomodated.

Typical crank vagueness and meaninglessness.

Too stupid to understand?

Potential infinity is used to enumerate the fractions, the algebraics,
the prime numbers. Very different multitudes. But the potentially
infinite sequence of natural numbers is very elastic. Example: After enumerating the positive fractions it can also absorb all negative
fractions.

Actual infinity is used to construct Cantors list which is too rigid to
absorb even one further number, namely the antidiagonal number.

Of course the leading liars don't like to understand that.

Regards, WM


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 23:08 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential
infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

I understand that these deeper notions are prohibited by the leading swindlers because they know that this knowledge disproves set theory.

This is a newsgroup for mathematics, not conspiracy theories.

Potential infinity: Hilbert's hotel. Every guest can be accomodated.
Actual infinity: Cantor's diagonal argument. The antidiagonal cannot be
accomodated.

Typical crank vagueness and meaninglessness.

Too stupid to understand?

No, too stupid to express yourself clearly.

Potential infinity is used to enumerate the fractions, the algebraics,
the prime numbers. Very different multitudes. But the potentially
infinite sequence of natural numbers is very elastic. Example: After enumerating the positive fractions it can also absorb all negative fractions.

Not really. There is no "after" enumerating the positive fractions.
They are infinite. Note the German word for infinite, "unendlich",
literally "without end".

"Absorb" is a vague crank word with no meaning in mathematics.

Actual infinity is used to construct Cantors list which is too rigid to absorb even one further number, namely the antidiagonal number.

"Actual infinity", whatever that might mean, is in no way "used" to
construct Cantor's non-existent list. Lists don't "absorb" numbers,
whatever that might mean. Cantor showed that any such purported
complete list was in fact not complete, by constructing a number not in
the list.

Of course the leading liars don't like to understand that.

The lack of understanding is your thing. Mathematicians aren't liars in
their own field. They just have a more developed knowledge of truth
than you do.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 23:08 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C).

Cardinality is nonsense. It shows a countable result when potential >>>>>> infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

I understand that these deeper notions are prohibited by the leading
swindlers because they know that this knowledge disproves set theory.

This is a newsgroup for mathematics, not conspiracy theories.

Mathematics unfortunately has becomen degraded this way. It is
impossible that the brighter leaders don't understand the mistakes of
set theory. Even the avarage mathematician knows that every indiviudelly chosen number has more successors than predecessors, such that at least
half of all natural numbers cannot be applied individuzally. But kind of Freudian repression prevents most to understand that the whole set rao can only be manipiulated collectively. Most numbers are dark such that Dedekin-Cantor's way of enumerating the algebraics must break down.

Potential infinity is used to enumerate the fractions, the algebraics,
the prime numbers. Very different multitudes. But the potentially
infinite sequence of natural numbers is very elastic. Example: After
enumerating the positive fractions it can also absorb all negative
fractions.

Not really. There is no "after" enumerating the positive fractions.

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Actual infinity is used to construct Cantors list which is too rigid to
absorb even one further number, namely the antidiagonal number.

"Actual infinity", whatever that might mean, is in no way "used" to
construct Cantor's non-existent list.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Lists don't "absorb" numbers,
whatever that might mean.

Hilbert's hotel can absorb another guest.

Cantor showed that any such purported
complete list was in fact not complete, by constructing a number not in
the list.

He claims that the enumeration is complete, and no new guest can be
absorbed.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 23:08 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 22.02.2026 um 21:32 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 21.02.2026 um 17:11 schrieb Ross Finlayson:

Cardinality establishes a transitive inequality among sets,
where "cardinals" themselves as equivalence classes of sets
having any transitive bijective relation, are, besides zero,
rather too large to be sets in ordinary set theories like ZF(C). >>>>>>> Cardinality is nonsense. It shows a countable result when potential >>>>>>> infinity is used but is greater when actual infinity is used.

More like "potential infinity" and "actual infinity" are nonsense.

Not everything you cannot comprehend is nonsense.

I understand that "potential infinity" and "actual infinity" are not
parts of modern mathematics. You appear not to.

I understand that these deeper notions are prohibited by the leading
swindlers because they know that this knowledge disproves set theory.

This is a newsgroup for mathematics, not conspiracy theories.

Mathematics unfortunately has become degraded this way. It is
impossible that the brighter leaders don't understand the mistakes of
set theory.

They will do, for what few mistakes there are. They also understand that
the supposed mistakes asserted by cranks are mostly nothing of the kind.

Even the average mathematician knows that every individually
chosen number has more successors than predecessors, ....

Of course.

.... such that at least half of all natural numbers cannot be applied individually.

They also know that operators are applied, not numbers.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must
break down.

Mathematicians don't think in these terms. They stick to mathematics.
"Dark numbers", as you "define" them have been proven on this newsgroup
not to exist.

Potential infinity is used to enumerate the fractions, the algebraics,
the prime numbers. Very different multitudes. But the potentially
infinite sequence of natural numbers is very elastic. Example: After
enumerating the positive fractions it can also absorb all negative
fractions.

Not really. There is no "after" enumerating the positive fractions.

Cantor enumerates the positive fractions. Then inserts each negative fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so. I have a degree in mathematics. You don't.

Actual infinity is used to construct Cantors list which is too rigid to
absorb even one further number, namely the antidiagonal number.

"Actual infinity", whatever that might mean, is in no way "used" to
construct Cantor's non-existent list.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't
come into it.

Lists don't "absorb" numbers, whatever that might mean.

Hilbert's hotel can absorb another guest.

A towel can absorb spilt liquids. At this point we're not discussing
maths, we're discussing correct English usage.

Cantor showed that any such purported complete list was in fact not
complete, by constructing a number not in the list.

He claims that the enumeration is complete, and no new guest can be absorbed.

You've completely lost the context, here.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 23.02.2026 um 18:16 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:

Even the average mathematician knows that every individually
chosen number has more successors than predecessors, ....

Of course.

.... such that at least half of all natural numbers cannot be applied
individually.

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must
break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so.

But you have forgotten all that?

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't
come into it.

Forgotten also that difference? The natural numbers can absorb more
fractions but not the diagonal number.

Regards, WM>

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/23/2026 09:27 AM, WM wrote:

Am 23.02.2026 um 18:16 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:

Even the average mathematician knows that every individually
chosen number has more successors than predecessors, ....

Of course.

.... such that at least half of all natural numbers cannot be applied
individually.

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must
break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so.

But you have forgotten all that?

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't
come into it.

Forgotten also that difference? The natural numbers can absorb more
fractions but not the diagonal number.

Regards, WM>

Actually duBois-Reymond discovered arguments for uncountability,
which were arguably simply extensions of considerations of
the Archimedean field, then Galileo's "paradox" relating the
integers and squares is pretty simple, too, courtesy Zeno.


That there are multiple models of continuous domains and
corresponding multiple laws of large numbers about the
existence of limit theorems vis-a-vis the inexistence of
a standard model of integers, makes for that a lot of what
were "uniqueness results" are "distinctness results", including
for example some of the results of the super-classical developments
of Fourier analysis, for which we can thank Dirichlet and Heine.

And Poincare, ....


It's pretty simple these days to make
screaming sock-puppet howler-trolls
of the retro-finitist variety.


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 18:16 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:
Even the average mathematician knows that every individually
chosen number has more successors than predecessors, ....

Of course.

.... such that at least half of all natural numbers cannot be applied
individually.

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

You just aren't familiar with mathematical usage. There is no meaning
of "apply" when the thing being applied is supposedly a set.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must
break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

No, you're mistaking their thought patterns for your own
misunderstanding of them.

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so.

But you have forgotten all that?

I've forgotten more about mathematics than you ever knew. The bit you
snipped, that I'm a graduate mathematician whereas you're not, you
simply have no reply for.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't
come into it.

Forgotten also that difference? The natural numbers can absorb more fractions but not the diagonal number.

That's meaningless crank talk. The set of natural numbers doesn't
"absorb" anything. Neither does an element of that set. To say so is a category error. A towel will absorb spilt fluid, but a set just
doesn't.
And as you ought to be aware, Cantor's diagonal number doesn't exist.
It is a purely hypothetical construct posited on the existence of a
complete list of real numbers.

Regards, WM>

--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:

So far that is only *your* claim, not Cantor's. You have not provided any >> evidence to support your claim.

I told you once and I told you twice
But you never listen to my advice
Well, this is the last time:
[Cantor, Collected Works, p. 132]

Irrelevant.

And, in fact, after you insisted that
Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence, which -- because of that description which is based on sequences of numbers that are coprime --
*by constrast to your claim* does NOT contain any duplicate rational numbers *from the outset* can be *constructed*, which *differs* from the one you
give, and *then* he suggests that there *should be* a formula and
*implicitly* asks Lipschitz for one.

Your problem is that you are unable to read *comprehensively*, and then you fill the gaps in your understanding with fantasies and conspiracy theories. Like a crackpot does.
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 23.02.2026 um 18:16 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so. I have a degree in mathematics. You don't.

I have been appointed by a committee of experts to give math lessons at
a university (Technische Hochschule Augsburg). At N|+rnberg I have won
the second place (with no doubt a mistake of the committee).

Some need to climb steps, others fly to the top.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 23.02.2026 um 23:41 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:

So far that is only *your* claim, not Cantor's. You have not provided any >>> evidence to support your claim.

I told you once and I told you twice
But you never listen to my advice
Well, this is the last time:
[Cantor, Collected Works, p. 132]

Irrelevant.

Chuckle.

And, in fact, after you insisted that
Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals
where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence,

But only without formula. The fractions have to be selected by hand like
the sequence of the algebraics.

there *should be* a formula and> *implicitly* asks Lipschitz for one.

He asks explicitly. Why? He anticipated that Fritsche could declare his
effort to be a supertask.

Regards, WM

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 23.02.2026 um 22:10 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

You just aren't familiar with mathematical usage. There is no meaning
of "apply" when the thing being applied is supposedly a set.

Enumerating is done applying natural numbers as indices.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must >>>> break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

No,

Yes, try to learn a bit at least. It is boring to have to repeat the
basics again and again!

Es lassen sich alsdann die Zahlen des Inbegriffes (NU+), d. h. s|nmtliche algebraischen reellen Zahlen folgenderma|fen anordnen: man nehme als
erste Zahl NU+1 die eine Zahl mit der H||he N = 1; lasse auf sie, der Gr|||fe nach steigend, die NU-(2) = 2 algebraischen reellen Zahlen der H||he N = 2 folgen, bezeichne sie mit NU+2, NU+3; an diese m||gen sich die NU-(3) = 4 Zahlen mit der H||he N = 3, ihrer Gr|||fe nach aufsteigend, anschlie|fen;

Do you see Zahl and Zahlen again and again!

I'm a graduate mathematician whereas you're

a professor having written five books about mathematics, one of them
with seven editions, one with four editions, published by De Gruyter.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't >>> come into it.

Invariability of sets is the basic of set theory.

And as you ought to be aware, Cantor's diagonal number doesn't exist.

True.

It is a purely hypothetical construct posited on the existence of a
complete list of real numbers.

Enumerated by the complete set of natural numbers!

Regards, WM>


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 22:10 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

You just aren't familiar with mathematical usage. There is no meaning
of "apply" when the thing being applied is supposedly a set.

Enumerating is done applying natural numbers as indices.

You must have mistranslated from German. You cannot "apply" a number.

But kind of Freudian repression prevents most to understand that the >>>>> whole set rao can only be manipulated collectively. Most numbers are >>>>> dark such that Dedekind-Cantor's way of enumerating the algebraics must >>>>> break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

No,

Yes, try to learn a bit at least. It is boring to have to repeat the
basics again and again!
Es lassen sich alsdann die Zahlen des Inbegriffes (NU+), d. h. s|nmtliche algebraischen reellen Zahlen folgenderma|fen anordnen: man nehme als
erste Zahl NU+1 die eine Zahl mit der H||he N = 1; lasse auf sie, der Gr|||fe
nach steigend, die NU-(2) = 2 algebraischen reellen Zahlen der H||he N = 2 folgen, bezeichne sie mit NU+2, NU+3; an diese m||gen sich die NU-(3) = 4 Zahlen mit der H||he N = 3, ihrer Gr|||fe nach aufsteigend, anschlie|fen;

This is an English language group. You should have translated the above.
But it is clear that the two mathematicians were not thinking in terms of
"dark numbers" or of their enumeration breaking down in any way. Nor
were they thinking about "manipulating the set N only collectively",
whatever that might mean.

Do you see Zahl and Zahlen again and again!

Yes. The writers are trying to describe a method of enumerating the
algebraic numbers, like you at first said. It looks to me like they
didn't manage it here, but I'm not sure about this.

I'm a graduate mathematician whereas you're

a professor having written five books about mathematics, one of them
with seven editions, one with four editions, published by De Gruyter.

Yes, I'm aware of these. They're controversial, to put it mildly. But
you don't have a degree in maths, and that lack has led you to wild and wondrous falsehoods, lacking, as you do, a fundamental grounding in the subject.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't >>>> come into it.

Invariability of sets is the basic of set theory.

What's that got to do with completeness?

And as you ought to be aware, Cantor's diagonal number doesn't exist.

True.

It is a purely hypothetical construct posited on the existence of a
complete list of real numbers.

Enumerated by the complete set of natural numbers!
Regards, WM>

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

wm wrote:

Am 23.02.2026 um 23:41 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:>>>> And, in fact, after you insisted that

Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals >>> where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence,

But only without formula.

He presents a bijective map of the natural numbers. That is enough.

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact the sequence, of numbers that are coprime to and less than a natural number.

there *should be* a formula and> *implicitly* asks Lipschitz for one.

He asks explicitly.

Cite the part of the letter that I quoted where you think that he asks explicitly. (You will not be able to because he simply does not do that.)
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.02.2026 um 13:52 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 22:10 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

You just aren't familiar with mathematical usage. There is no meaning
of "apply" when the thing being applied is supposedly a set.

Enumerating is done applying natural numbers as indices.

You must have mistranslated from German. You cannot "apply" a number.

Of course I ca, It is a tool for indexing.

But kind of Freudian repression prevents most to understand that the >>>>>> whole set rao can only be manipulated collectively. Most numbers are >>>>>> dark such that Dedekind-Cantor's way of enumerating the algebraics must >>>>>> break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

No,

Yes, try to learn a bit at least. It is boring to have to repeat the
basics again and again!

Es lassen sich alsdann die Zahlen des Inbegriffes (NU+), d. h. s|nmtliche
algebraischen reellen Zahlen folgenderma|fen anordnen: man nehme als
erste Zahl NU+1 die eine Zahl mit der H||he N = 1; lasse auf sie, der Gr|||fe
nach steigend, die NU-(2) = 2 algebraischen reellen Zahlen der H||he N = 2 >> folgen, bezeichne sie mit NU+2, NU+3; an diese m||gen sich die NU-(3) = 4
Zahlen mit der H||he N = 3, ihrer Gr|||fe nach aufsteigend, anschlie|fen;

This is an English language group. You should have translated the above.

You would accjse me of mistranlating. Further a citizen of N|+rnberg
should understand some German.

But it is clear that the two mathematicians were not thinking in terms of "dark numbers" or of their enumeration breaking down in any way.

They are using numbers for enumerating purposes. That was the question.

Do you see Zahl and Zahlen again and again!

Yes. The writers are trying to describe a method of enumerating the algebraic numbers, like you at first said. It looks to me like they
didn't manage it here, but I'm not sure about this.

They did!

I'm a graduate mathematician whereas you're

a professor having written five books about mathematics, one of them
with seven editions, one with four editions, published by De Gruyter.

Yes, I'm aware of these. They're controversial, to put it mildly.

A German liar has uttered his hate.

But
you don't have a degree in maths,

Did Pythagoras have a degree? Archimedes, Euclid?

Invariability of sets is the basic of set theory.

What's that got to do with completeness?

Completeness and invariability lead to a greatest element of linearly
ordered sets.

And as you ought to be aware, Cantor's diagonal number doesn't exist.

True.

It is a purely hypothetical construct posited on the existence of a
complete list of real numbers.

Enumerated by the complete set of natural numbers!

Regards, WM>


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

Am 23.02.2026 um 23:41 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:>>>> And, in fact, after you insisted that

Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals >>>> where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence,

But only without formula.

He presents a bijective map of the natural numbers. That is enough.

Selected by hand.

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact the sequence, of numbers that are coprime to and less than a natural number.

No.

> there *should be* a formula and> *implicitly* asks Lipschitz for one.

He asks explicitly.

Cite the part of the letter that I quoted where you think that he asks explicitly. (You will not be able to because he simply does not do that.)

Liesse sich nicht mit den Mitteln der analytischen Zahlentheorie (Ausdrucksweise von Mertens) ein analytischer Ausdruck f|+r die Function F(NU<) finden? Etwa durch ein bestimmtes Integral, welches NU< als Parameter enth|nlt?

Regards, WM

--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

Am 23.02.2026 um 23:41 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:>>>> And, in fact, after you insisted that

Cantor had used it in a letter that he wrote to Lipschitz in 1883,

No, there he *asked* for a formula to describe his sequence of rationals >>>>> where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence,

But only without formula.

He presents a bijective map of the natural numbers. That is enough.

Selected by hand.

*facepalm*

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact >> the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code
that does what can be done more tediously by manual calculation as well:

--[math/integer.js]-----------------------------------------------------

jsx.math.integer = {
/**
* Return the common divisors of two or more integers.
*
* @params {int}
* @returns {Array<int>} Common divisors of the arguments
*/
commonDivisors: function () {
const args = Array
.from(arguments)
.sort((a, b) => (a < b) ? -1 : ((a > b) ? 1 : 0));

const min = args[0];

const divisorCandidates = Array
.from({ length: min + 1 }, (_, x) => x)
.filter(x => (x > 0) && (x <= min));

const result = divisorCandidates
.filter(divisor => args.every(arg => (arg % divisor === 0)));

return result;
},

// [...]

/**
* Return the positive integers that are relative prime to
* and smaller than a positive integer.
*
* @param {int} n
* @returns {Array<int>}
*/
relPrimes: function (n) {
const _commonDivisors = jsx.math.integer.commonDivisors;
const result = Array
.from({length: n}, (_, x) => x)
.filter(i => {
const commonDivs = _commonDivisors(i, n);
return (commonDivs.length === 1 && commonDivs[0] === 1);
});

return result;
}
};

------------------------------------------------------------------------

[to be included in my free software library, JSX [1], at some later time.

[1] <https://github.com/PointedEars/JSX>]

In order to generate Cantor's sequence, then, you just have to call
relPrimes() repeatedly while increasing the argument by 1 in each step, starting with 2.

Counting then the elements of the sequence of generated sequences instead of counting by sections, which is what Cantor describes as the final step, is child's play.

> there *should be* a formula and> *implicitly* asks Lipschitz for one. >>>
He asks explicitly.

Cite the part of the letter

I should have said: "_Quote_ the part of the letter [...]" (it is a "false friend" particularly for a native speaker of German). But you understood correctly what I meant (proving once more that you are a native speaker of German, so it is not a language issue; you just can't read comprehensively).

that I quoted where you think that he asks explicitly. (You will not be able to
because he simply does not do that.)

Liesse sich nicht mit den Mitteln der analytischen Zahlentheorie (Ausdrucksweise von Mertens) ein analytischer Ausdruck f|+r die Function F(NU<) finden? Etwa durch ein bestimmtes Integral, welches NU< als Parameter enth|nlt?

(sic)

[Learn to quote: A quotation is enclosed in quotation marks or each line is preceded by a prefix. Also, get yourself a proper newsreader that does not mutilate properly declared and encoded Unicode characters: <https://thunderbird.net/>]

That is not an explicit *request*, but an *implicit* _question_. Look up
what "explicit" and "implicit" mean, and what the difference between a
request and a question is.

For example, an explicit request would be "Please let me know if you know a
way to ...". But Cantor merely asks rhetorical questions: "Wouldn't it be possible to ...? For example, using ...?"

(q.e.d.)
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 2/25/2026 3:36 PM, Thomas 'PointedEars' Lahn wrote:
[...]

Side note, the cantor pairing can be used for fun things... Music,
hashes? Here is an example of some fun with MIDI:

https://youtu.be/XkwgJt5bxKI
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

Am 23.02.2026 um 23:41 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 22.02.2026 um 22:52 schrieb Thomas 'PointedEars' Lahn:>>>> And, in fact, after you insisted that

Cantor had used it in a letter that he wrote to Lipschitz in 1883, >>>>>>

No, there he *asked* for a formula to describe his sequence of rationals >>>>>> where the multiple appearances had been deleted by hand.

No, he gives a description of how his *actual* sequence,

But only without formula.

He presents a bijective map of the natural numbers. That is enough.

Selected by hand.

*facepalm*

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code that does what can be done more tediously by manual calculation as well:

Of course. Here the computer does it. But it is not a closed formula.

> there *should be* a formula and> *implicitly* asks Lipschitz for one. >>>>
He asks explicitly.

Cite the part of the letter

I should have said: "_Quote_ the part of the letter [...]" (it is a "false friend" particularly for a native speaker of German). But you understood correctly what I meant (proving once more that you are a native speaker of German, so it is not a language issue; you just can't read comprehensively).

that I quoted where you think that he asks explicitly. (You will not be able to
because he simply does not do that.)

Liesse sich nicht mit den Mitteln der analytischen Zahlentheorie
(Ausdrucksweise von Mertens) ein analytischer Ausdruck f|+r die Function
F(NU<) finden? Etwa durch ein bestimmtes Integral, welches NU< als Parameter >> enth|nlt?

That is not an explicit *request*,

It is.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/23/2026 01:10 PM, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 18:16 schrieb Alan Mackenzie:

wm <wolfgang.mueckenheim@tha.de> wrote:

Am 23.02.2026 um 14:45 schrieb Alan Mackenzie:

Even the average mathematician knows that every individually
chosen number has more successors than predecessors, ....

Of course.

.... such that at least half of all natural numbers cannot be applied
individually.

They also know that operators are applied, not numbers.

For enumerating the algebraic numbers natural numbers are applied.

You just aren't familiar with mathematical usage. There is no meaning
of "apply" when the thing being applied is supposedly a set.

But kind of Freudian repression prevents most to understand that the
whole set rao can only be manipulated collectively. Most numbers are
dark such that Dedekind-Cantor's way of enumerating the algebraics must >>>> break down.

Mathematicians don't think in these terms.

Dedekind and Cantor did.

No, you're mistaking their thought patterns for your own
misunderstanding of them.

Cantor enumerates the positive fractions. Then inserts each negative
fraction behind the positive fraction. Please try to learn the stuff
before you play the expert.

Have done so.

But you have forgotten all that?

I've forgotten more about mathematics than you ever knew. The bit you snipped, that I'm a graduate mathematician whereas you're not, you
simply have no reply for.

Please try to learn the stuff before you play the expert.
The list is considered complete by using all natural numbers for
enumerating the entries.

Or, more mathematically expressed, a bijection is created between the
natural numbers and the rational numbers, or whatever countable set
you're thinking about. Completeness, whatever you mean by that, doesn't >>> come into it.

Forgotten also that difference? The natural numbers can absorb more
fractions but not the diagonal number.

That's meaningless crank talk. The set of natural numbers doesn't
"absorb" anything. Neither does an element of that set. To say so is a category error. A towel will absorb spilt fluid, but a set just
doesn't.

And as you ought to be aware, Cantor's diagonal number doesn't exist.
It is a purely hypothetical construct posited on the existence of a
complete list of real numbers.

Regards, WM>

With the natural/unit equivalency function,
not a Cartesian function,
the antidiagonal is always at the end,
which always exists.

Not being a Cartesian function then is simply
a neat, brief result preventing otherwise
contradiction about Cantor-Schroeder-Bernstein
theorem, which is quite a long way to say that
Cartesian functions compose capriciously,
that thusly besides constructively having the
extent, density, completeness, and measure to
make ran(EF) a continuous domain, that furthermore
it's also not contradicted by a wider account of
ordinary set theory, which is quite thoroughly
established as a candidate "Foundations of Mathematics",
being talk about a simple theory-of-one-relation.


Cardinals are equivalence classes of sets relating
by transitive composability of bijective functions.
Cardinals besides the trivial case of the empty set
being defined as unique, are "too large" to be sets.

In an ordinary set theory, ....

It's easy to brickbat Dumb-Em, where it persists
in being a sort of howler crank troll.


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code >> that does what can be done more tediously by manual calculation as well:

Of course. Here the computer does it. But it is not a closed formula.

Irrelevant. Nobody but you claimed that there was one, and nobody but you
is claiming that there has to be one for the rationals to be countable.
You are wrong.

["Liesse sich nicht mit den Mitteln der analytischen Zahlentheorie
(Ausdrucksweise von Mertens) ein analytischer Ausdruck f|+r die Function >>> F(NU<) finden? Etwa durch ein bestimmtes Integral, welches NU< als Parameter
enth|nlt?"]

That is not an explicit *request*,

It is.

No, it is not. You are a hopeless case, unable to read comprehensively.
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/26/2026 01:46 PM, Thomas 'PointedEars' Lahn wrote:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code >>> that does what can be done more tediously by manual calculation as well:

Of course. Here the computer does it. But it is not a closed formula.

Irrelevant. Nobody but you claimed that there was one, and nobody but you
is claiming that there has to be one for the rationals to be countable.
You are wrong.

["Liesse sich nicht mit den Mitteln der analytischen Zahlentheorie
(Ausdrucksweise von Mertens) ein analytischer Ausdruck f|+r die Function >>>> F(NU<) finden? Etwa durch ein bestimmtes Integral, welches NU< als Parameter
enth|nlt?"]

That is not an explicit *request*,

It is.

No, it is not. You are a hopeless case, unable to read comprehensively.

PointedEars, maybe you didn't notice,
but I'm the Elephant in this room.

The "Relephant".


--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 26.02.2026 um 22:46 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code >>> that does what can be done more tediously by manual calculation as well:

Of course. Here the computer does it. But it is not a closed formula.

Irrelevant. Nobody but you claimed that there was one, and nobody but you
is claiming that there has to be one for the rationals to be countable.

I do not claim that. Moebius did. So:

You are wrong.

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence

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

Its terms can be represented by matrices. When we attach all indeXes k =
1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1
and indicate missing indexes by hOles O, then we get the matrix M(0) as starting position:

XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
... ... ... ...
M(0) M(2) M(3) M(4) ...

M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index 2
from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3. Successively all fractions of the sequence get indexed. In the limit,
denoted by M(reR), we see no fraction without index remaining. Note that
the only difference to Cantor's enumeration is that Cantor does not
render account for the source of the indices.

Every X, representing the index k, when taken from its present fraction
m/n, is replaced by the O taken from the fraction to be indexed by this
k. Its last carrier m/n will be indexed later by another index.
Important is that, when continuing, no O can leave the matrix as long as
any index X blocks the only possible drain, i.e., the first column. And
if leaving, where should it settle?

As long as indexes are in the drain, no O has left. The presence of all
O indicates that almost all fractions are not indexed. And after all
indexes have been issued and the drain has become free, no indexes are available which could index the remaining matrix elements, yet covered by O.

It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized. Lossless transpositions cannot suffer losses. The limit matrix M(reR) only shows what should have happened when
all fractions were indexed. Logic proves that this cannot have happened
by exchanges. The only explanation for finally seeing M(reR) is that there
are invisible matrix positions, existing already at the start. Obviously
by exchanging O and X no O can leave the matrix, but the O can disappear
by moving without end, from visible to invisible positions.

The number of not indexed fractions remains |rao|*(|rao|-1) for all
definable terms of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3,
3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
Hence |rao|*(|rao|-1) fractions cannot be indexed by definable indices.

Regards, WM
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

WM wrote:

Am 26.02.2026 um 22:46 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code >>>> that does what can be done more tediously by manual calculation as well: >>>

Of course. Here the computer does it. But it is not a closed formula.

Irrelevant. Nobody but you claimed that there was one, and nobody but you >> is claiming that there has to be one for the rationals to be countable.

I do not claim that.

Yes, you do. You have done it before, and you are doing it here again.

You are wrong.

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

:-D

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

Again, that is your claim, not Cantor's.

Intelligent life is characterized by learning from its mistakes. Given that you are making the same mistake over and over again even when it has been carefully explained to you, you are not exactly in a good position to even recognize an "intelligent mathematician".
--
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--- Synchronet 3.21b-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/27/2026 03:39 PM, Thomas 'PointedEars' Lahn wrote:

WM wrote:

Am 26.02.2026 um 22:46 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of the algebraics.

No, the fractions can be *computed*. You can *compute* the set, and in fact
the sequence, of numbers that are coprime to and less than a natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible source code
that does what can be done more tediously by manual calculation as well: >>>>

Of course. Here the computer does it. But it is not a closed formula.

Irrelevant. Nobody but you claimed that there was one, and nobody but you >>> is claiming that there has to be one for the rationals to be countable.

I do not claim that.

Yes, you do. You have done it before, and you are doing it here again.

You are wrong.

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

:-D

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

Again, that is your claim, not Cantor's.

Intelligent life is characterized by learning from its mistakes. Given that you are making the same mistake over and over again even when it has been carefully explained to you, you are not exactly in a good position to even recognize an "intelligent mathematician".

Of course, given the "ordered field" the Pythagorean before the
complete ordered field, and not having members of the "complete
ordered field" not in the rational numbers for the missing elements
to fall out: one may write "nested intervals" or "the antidiagonal"
for the rationals, given the reals don't already exist of otherwise
for people who call the rationals "complete" like Pythagoreans,
for example as once slipped out of "Metamath" with regards to "ruc"
"rationals uncountable", or any other system of inference that's not
thorough.


--- Synchronet 3.21d-Linux NewsLink 1.2

From Newsgroup: sci.math

On 02/27/2026 08:03 PM, Ross Finlayson wrote:

On 02/27/2026 03:39 PM, Thomas 'PointedEars' Lahn wrote:

WM wrote:

Am 26.02.2026 um 22:46 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of >>>>>>>>> the algebraics.

No, the fractions can be *computed*. You can *compute* the set, >>>>>>>> and in fact
the sequence, of numbers that are coprime to and less than a
natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible
source code
that does what can be done more tediously by manual calculation as >>>>>> well:

Of course. Here the computer does it. But it is not a closed formula. >>>>

Irrelevant. Nobody but you claimed that there was one, and nobody
but you
is claiming that there has to be one for the rationals to be countable. >>>

I do not claim that.

Yes, you do. You have done it before, and you are doing it here again.

You are wrong.

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

:-D

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

Again, that is your claim, not Cantor's.

Intelligent life is characterized by learning from its mistakes.
Given that
you are making the same mistake over and over again even when it has been
carefully explained to you, you are not exactly in a good position to
even
recognize an "intelligent mathematician".

Of course, given the "ordered field" the Pythagorean before the
complete ordered field, and not having members of the "complete
ordered field" not in the rational numbers for the missing elements
to fall out: one may write "nested intervals" or "the antidiagonal"
for the rationals, given the reals don't already exist of otherwise
for people who call the rationals "complete" like Pythagoreans,
for example as once slipped out of "Metamath" with regards to "ruc" "rationals uncountable", or any other system of inference that's not thorough.

Anytime you have to add a "rescue lemma", whether that's adding an
axiom or picking a branch, in mathematical proof, it's apparent that
there's either an "opening" ("perestroika", "catastrophe") and also
"the super-classical".

For example, "Cohen's forcing above all ZF(C)" has otherwise
it's broken at least two ways.

Then, an account of paradox-free reason about the universe of
mathematical objects demands of itself to be quite: thorough.

Singularities in a singularity theory are just branches in
a multiplicity theory. Otherwise you've got the engineer's
disease, a sort of aphasia generously or just ignorance.


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From Newsgroup: sci.math

Am 28.02.2026 um 00:39 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

Again, that is your claim, not Cantor's.

Es hat n|nmlich die Funktion NU4 = NU! + (NU!NCaNCaNC2NCaNU<NCa- 1)(NU!NCaNC2NCaNU<NCa- 2)/2, wie
leicht zu zeigen, die bemerkenswerte Eigenschaft, da|f sie alle positiven ganzen Zahlen und jede nur einmal darstellt, wenn in ihr NU!NCa undNCaNU<NCaunabh|nngig voneinander ebenfalls jeden positiven, ganzzahligen Wert erhalten [1]. [Cantor, Collected Works p. 132]

Gru|f, WM
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On 02/27/2026 08:03 PM, Ross Finlayson wrote:

On 02/27/2026 03:39 PM, Thomas 'PointedEars' Lahn wrote:

WM wrote:

Am 26.02.2026 um 22:46 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 26.02.2026 um 00:36 schrieb Thomas 'PointedEars' Lahn:

WM wrote:

Am 25.02.2026 um 18:10 schrieb Thomas 'PointedEars' Lahn:

wm wrote:

The fractions have to be selected by hand like the sequence of >>>>>>>>> the algebraics.

No, the fractions can be *computed*. You can *compute* the set, >>>>>>>> and in fact
the sequence, of numbers that are coprime to and less than a
natural number.

No.

:-D

Here are the relevant parts of my ECMAScript Ed. 6+ compatible
source code
that does what can be done more tediously by manual calculation as >>>>>> well:

Of course. Here the computer does it. But it is not a closed formula. >>>>

Irrelevant. Nobody but you claimed that there was one, and nobody
but you
is claiming that there has to be one for the rationals to be countable. >>>

I do not claim that.

Yes, you do. You have done it before, and you are doing it here again.

You are wrong.

By the way, closed formula or not: The fractioons are not countable.
Every intelligent mathematician can understand my proof:

:-D

According to Cantor all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...

can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m

Again, that is your claim, not Cantor's.

Intelligent life is characterized by learning from its mistakes.
Given that
you are making the same mistake over and over again even when it has been
carefully explained to you, you are not exactly in a good position to
even
recognize an "intelligent mathematician".

Of course, given the "ordered field" the Pythagorean before the
complete ordered field, and not having members of the "complete
ordered field" not in the rational numbers for the missing elements
to fall out: one may write "nested intervals" or "the antidiagonal"
for the rationals, given the reals don't already exist of otherwise
for people who call the rationals "complete" like Pythagoreans,
for example as once slipped out of "Metamath" with regards to "ruc" "rationals uncountable", or any other system of inference that's not thorough.

So, making it so that mathematics has relations instead of
contradictions about infinitary reasoning involves a sufficiently
thorough account about the independence of arithmetic, at
least three laws of large numbers, at least three continuous
domains, at least three Cantor spaces, at least three probabilistic
limit theorems, and otherwise an "Atlas of Mathematical Independence".


The usual narrative's account only has one when there are at
least three, so it's not a suitable candidate for Foundations.


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