On 1/26/2023 5:28 PM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 26. Januar 2023 um 20:52:38 UTC+1:
>> On 1/26/2023 5:11 AM, WM wrote:
>>> Jim Burns schrieb am Mittwoch,
>>> 25. Januar 2023 um 23:54:40 UTC+1:

>>>> For each finite ordinal, dark or visible,
>>>> for each split between 0 and that ordinal
>>>> some i,j is last-before,first-after
>>>> and j = i+1
>>>
>>> That is your unfounded claim,
>>> obtained from the first part of
>>> the sequenve of real numbers.
>>
>> No,
>> that claim is well-supported by it being
>> _what we mean_ by "finite ordinal".
>
> But there are others which you don't mean

Yes.
Exactly.
There are others which we don't mean.

We make descriptive claims of _the ones we mean_
Those claims are not-possibly-false
when referring to _the ones we mean_

We augment those claims with visibly
not-first-possibly-false claims which also
refer to _the ones we mean_

In that way, we assemble a sequence of only
not-possibly-false claims about _the ones we mean_

There are others which we don't mean.

The reason that
the claims about _the ones that we mean_
are not-possibly-false is that they aren't
claims about _the ones we don't mean_

> But there are others which you don't mean
> because you have not yet seen them.

No,
if that were the reason we don't mean them,
then _the ones we mean_ would never be what
we have not yet seen.

That is often not the case. Arriving at
not-possibly-false claims about what we have
not yet seen is the chief stock-in-trade
of mathematics.

William schrieb am Freitag, 27. Januar 2023 um 00:43:02 UTC+1:
> On Thursday, January 26, 2023 at 6:25:46 PM UTC-4, WM wrote:
>
>
> > Please follow this simple example: Move the cursor from 1 to 0 and note all unit fractions.
> This cannot be done.

The cursor can move, but you are right that not all unit fractions can be noted. (If however Cantor's bijections should be accepted, then all must be noted: "The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

> There are an infinite number of U_P. You either pass an infinite number of elements of U_P and get to zero or there is an element of U_P you do not pass.

If they are there, then they have been passed.

> There is no third choice. There is no last Peano number.

There is a point, namely zero, such that no unit fraction will follow. Hence there must have been a last one passed and a last one noted. Both cannot be identical because the last one noted has a finite number.

> Hence a Peano set cannot be finite.

The noted ones are finite. To prove this move from 0 to 1. The first one noted has a finite n.

> A Peano set does not change

When you go from 0 to 1, the set of noted unit fractions is arbitrarily large but finite.

> so it is not finite but unbounded. A Peano set has cardinality ℵo..

That is a blatantly counterfactual statement. I don't believe that you really believe in it.

Regards, WM

Jim Burns schrieb am Freitag, 27. Januar 2023 um 01:54:08 UTC+1:

> That is often not the case. Arriving at
> not-possibly-false claims about what we have
> not yet seen is the chief stock-in-trade
> of mathematics.

Then you know that all unit fractions have an ℵo-infinite distance (measured in unit fradtions) from zero. Then you will never be able to move a cursor from 1 to zero because at no point you will have to go less than ℵo points. What a pity.

Regards, WM

On Friday, January 27, 2023 at 11:32:40 AM UTC+1, WM wrote:
> Jim Burns schrieb am Freitag, 27. Januar 2023 um 01:54:08 UTC+1:
>
> > That is often not the case. Arriving at
> > not-possibly-false claims about what we have
> > not yet seen is the chief stock-in-trade
> > of mathematics.
> Then you know that all unit fractions have an ℵo-infinite distance (measured in unit fradtions) from zero. Then you will never be able to move a cursor from 1 to zero because at no point you will have to go less than ℵo points. What a pity.
>
> Regards, WM

So Achilles will never get to eat his turtle soup.
Maybe he shouldn't have measured the distance in "unit fradtions".

On Friday, January 27, 2023 at 5:09:44 AM UTC-4, WM wrote:

> > A Peano set has cardinality ℵo.
> That is a blatantly counterfactual statement. I don't believe that you really believe in it.
There are only three choices. Finite, finite but unbounded, cardinality ℵo. The first two are ruled out

On 1/27/2023 5:32 AM, WM wrote:
> Jim Burns schrieb am Freitag,
> 27. Januar 2023 um 01:54:08 UTC+1:

>>> But there are others which you don't mean
>>> because you have not yet seen them.
>>
>> No,
>> if that were the reason we don't mean them,
>> then _the ones we mean_ would never be what
>> we have not yet seen.

>> That is often not the case. Arriving at
>> not-possibly-false claims about what we have
>> not yet seen is the chief stock-in-trade
>> of mathematics.
>
> Then you know
> that all unit fractions

_The unit fractions we mean_ are in
a sequence ⟨ 1/1 1/2 1/3 ... ⟩ in which,
for each split,
some 1/j,1/j⁺⁺ is last-before,first-after.

> Then you know
> that all unit fractions
> have an ℵo-infinite distance
> (measured in unit fradtions) from zero.

Yes.

There is a match 1/k:1/k⁺⁺ between
all unit fractions after 1/j and
all unit fractions after 1/j⁺⁺

Therefore,
there is no _first_ unit fraction 1/j++
for which there _isn't_ a match between
all unit fractions after 1/j⁺⁺ and
all unit fractions after 1/1
but there _is_ a match between
all unit fractions after 1/j and
all unit fractions after 1/1

Therefore,
there is no unit fraction 1/j _at all_
for which there _isn't_ a match between
all unit fractions after 1/j and
all unit fractions after 1/1

Therefore,
for each unit fraction 1/j
there are ℵ₀ = |ℕ| unit fractions after 1/j

where we write
ℵ₀ = |ℕ| for the cardinality of
the unit fractions after 1/1

Because,
if F is a FISON, then F⊕⟨F⌝⁺⁺⟩ is a FISON too,
and _the ℕ we mean_ is the set of FISON-ends.

> Then you will never be able to move a cursor
> from 1 to zero

No,
Achilles does actually pass the tortoise.

You have given an incorrect description of
movements of a cursor, of Achilles.

What we do is
we describe one of _the ones we mean_
_correctly_

That _correct_ description gives us some
not-possibly-false claims about the ones we mean.

We augment those not-possibly-false claims
with visibly not-first-possibly-false claims.

Each claim in the augmented sequence is
a not-first-possibly-false claim
about the ones we mean.

Therefore,
each claim in the augmented sequence is
a not-possibly-false claim
about the ones we mean.

Our use of all-not-first-possibly-false sequences
_begins_ with a _correct_ description.
You didn't begin with a correct description.

> because
> at no point you will have to go
> less than ℵo points.
> What a pity.

For any movement between rational points
in their standard order,
there are ℵ₀ = |ℕ| points passed.

For the same points
in Cantor's order,
there are finitely-many points passed.

That is not because points disappear or appear.
They are the same points in two different orders.

It is because infinite is not
a reallyreallyreallyreallyreallyreally large
natural number.

And the reason behind that reason is that
F⊕⟨F⌝⁺⁺⟩ is a FISON too.

William schrieb am Freitag, 27. Januar 2023 um 18:42:44 UTC+1:
> On Friday, January 27, 2023 at 5:09:44 AM UTC-4, WM wrote:
>
>
> > > A Peano set has cardinality ℵo.
> > That is a blatantly counterfactual statement. I don't believe that you really believe in it.
> There are only three choices. Finite, finite but unbounded, cardinality ℵo. The first two are ruled out

Cardinality ℵo is ruled out for the collection of elements which have ℵo successors.
Find a Peano element that has less successors. Fail.

There is a war. There are two antagonistic forces. One is trying to extend the collection of defined natnumbers. The other is trying to keep the set of undefined natnumbers infinite. But note that there is only one ℵo available in cosecutively sharing ℕ.

Regards, WM

Jim Burns schrieb am Freitag, 27. Januar 2023 um 19:44:57 UTC+1:
> On 1/27/2023 5:32 AM, WM wrote:

> > Then you know
> > that all unit fractions
> > have an ℵo-infinite distance
> > (measured in unit fradtions) from zero.
> Yes.

And you cannot find any extending the interval (0, 1] to 0. But the interval is completely covered by the unit fractions.

> > at no point you will have to go
> > less than ℵo points.
> > What a pity.
> For any movement between rational points
> in their standard order,
> there are ℵ₀ = |ℕ| points passed.

Yes, and you cannot stop at one point which has a smaller distance from another point. That means the ℵ₀ points passed are dark. And if you half them, then the halves are dark. It's like quarks or magnets.

Regards, WM

On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:

> Find a Peano element that has less [than ℵo] successors. Fail.

Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.

On 1/27/2023 2:25 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 27. Januar 2023 um 19:44:57 UTC+1:
>> On 1/27/2023 5:32 AM, WM wrote:

>>> at no point you will have to go
>>> less than ℵo points.
>>> What a pity.
>>
>> For any movement between rational points
>> in their standard order,
>> there are ℵ₀ = |ℕ| points passed.
>
> Yes,
> and you cannot stop at one point
> which has a smaller distance from another point.

_No point exists_ which has a smaller
distance (cardinality-between) from another point.

For each rational p in open interval (a,b)
there is one and only one rational q in ℚ⁺
such that
(b-a)/(p-a) = q+1

For each rational q in ℚ⁺
there is one and only one rational p in (a,b)
such that
(b-a)/(p-a) = q+1

> That means the ℵ₀ points passed are dark.

No,
that means that infinity is not
a reallyreallyreallyreallyreallyreally large
natural number.

> And if you half them, then the halves are dark.
> It's like quarks or magnets.

Jim Burns schrieb am Freitag, 27. Januar 2023 um 23:23:43 UTC+1:
> On 1/27/2023 2:25 PM, WM wrote:

> >> For any movement between rational points
> >> in their standard order,
> >> there are ℵ₀ = |ℕ| points passed.

Why do you not stop before?
> >
> > Yes,
> > and you cannot stop at one point
> > which has a smaller distance from another point.
> _No point exists_ which has a smaller
> distance (cardinality-between) from another point.

You said that ℵ₀ points are passed before you can stop. Do they exist?

Regards, WM

On 1/27/2023 5:30 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 27. Januar 2023 um 23:23:43 UTC+1:
>> On 1/27/2023 2:25 PM, WM wrote:

>>>> For any movement between rational points
>>>> in their standard order,
>>>> there are ℵ₀ = |ℕ| points passed.
>
> Why do you not stop before?

You're asking
why is there no interval with finitely-many
rationals in it?

Because,
for open intervals (a,b) (c,d) of rationals,

for each p in (a,b)
one and only one q in (c,d) exists
such that (b-a)/(p-a) = (d-c)/(q-c)

and
for each q in (c,d)
one and only one p in (a,b)
such that (b-a)/(p-a) = (d-c)/(q-c)

>>> Yes,
>>> and you cannot stop at one point
>>> which has a smaller distance from another point.

>> _No point exists_ which has a smaller
>> distance (cardinality-between) from another point.
>
> You said that ℵ₀ points are passed
> before you can stop. Do they exist?

Let q be an element of ℚ⁺

n exists such that,
a sequence ⟨ 0 1 2 ... n ⟩ exists such that,
for each split between 0 and n
some i,i⁺⁺ is last-before,first-after
and
a sequence ⟨ 0 q 2q ... n ⟩ exists such that,
for each split between 0 and n
some p,p+q is last-before,first-after.

Does q exist?

William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
>
> > Find a Peano element that has less [than ℵo] successors. Fail.
>
> Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.

1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
3) Its size differs according to the effort of the observer.
4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.

Would you be so kind to note which step is wrong in your opinion?

Regards, WM

Jim Burns schrieb am Samstag, 28. Januar 2023 um 00:33:43 UTC+1:
> On 1/27/2023 5:30 PM, WM wrote:

> You're asking
> why is there no interval with finitely-many
> rationals in it?

Yes.
>
> Because,
> for open intervals (a,b) (c,d) of rationals,
>
> for each p in (a,b)
> one and only one q in (c,d) exists
> such that (b-a)/(p-a) = (d-c)/(q-c)

That is not explaining the reason of the fatcs.
>

> > You said that ℵ₀ points are passed
> > before you can stop. Do they exist?
> Let q be an element of ℚ⁺

No, the question was: Do they exist?

Please answer yes or no to the following four points without your usual waffle:

1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
3) Its size differs according to the effort of the observer.
4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.

Regards, WM

On Saturday, January 28, 2023 at 12:06:13 PM UTC+1, WM wrote:
> William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> >
> > > Find a Peano element that has less [than ℵo] successors. Fail..
> >
> > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
> 3) Its size differs according to the effort of the observer.
> 4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.
>
> Would you be so kind to note which step is wrong in your opinion?
>
> Regards, WM

2, 3, and 4 are the usual meaningless garble.
Define your terms.

JVR schrieb am Samstag, 28. Januar 2023 um 12:13:19 UTC+1:
> On Saturday, January 28, 2023 at 12:06:13 PM UTC+1, WM wrote:
> > William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> > >
> > > > Find a Peano element that has less [than ℵo] successors. Fail.
> > >
> > > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> > 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> > 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
> > 3) Its size differs according to the effort of the observer.
> > 4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.
> >
> > Would you be so kind to note which step is wrong in your opinion?

> 2, 3, and 4 are the usual meaningless garble.
> Define your terms.

Define a unit fraction such that the difference disappears. Then you will learn what defining means.

Regards, WM

On Saturday, January 28, 2023 at 12:36:05 PM UTC+1, WM wrote:
> JVR schrieb am Samstag, 28. Januar 2023 um 12:13:19 UTC+1:
> > On Saturday, January 28, 2023 at 12:06:13 PM UTC+1, WM wrote:
> > > William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > > > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> > > >
> > > > > Find a Peano element that has less [than ℵo] successors. Fail.
> > > >
> > > > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> > > 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> > > 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
> > > 3) Its size differs according to the effort of the observer.
> > > 4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.
> > >
> > > Would you be so kind to note which step is wrong in your opinion?
> > 2, 3, and 4 are the usual meaningless garble.
> > Define your terms.
> Define a unit fraction such that the difference disappears. Then you will learn what defining means.
>
> Regards, WM

It's ok, Mücke. You can keep making an ass of yourself here. Almost nobody will notice.
Usenet is the ideal playpen for people like you.

On 1/28/2023 6:11 AM, WM wrote:
> Jim Burns schrieb am Samstag,
> 28. Januar 2023 um 00:33:43 UTC+1:

>> You're asking
>> why is there no interval with
>> finitely-many rationals in it?
>
> Yes.

Because
each point in the whole ℚ⁺
can be matched to
its own point in part of ℚ⁺

Whatever part, whatever match,
that match shows that there are
infinitely-many points in ℚ⁺

A part of ℚ⁺ for which
each point in it can be matched to
its own point in the whole ℚ⁺
also has infinitely-many points.
Cantorianly,
the matchable part has the same number
of points as ℚ⁺ which is infinite.

Matchable parts of ℚ⁺
include
each open interval (a,b) with rational endpoints.

That is why no interval has
finitely-many rationals in it.

----
We can support that claim by starting with
a simple not-possibly-false claim about
a point p in (a,b)
| p is a point in (a,b)

We display the procedure that produces
the point q in ℚ⁺ which is uniquely matched to p
Similar triangles can do it, geometrically.

And then,
we claim, for a point p₂ in (a,b)
| p₂ is a point in (a,b)

The procedure which produces q for p
produces q₂ for p₂

Show that,
if q₂ = q, then p₂ = p
No _different_ point p₂ has q, only p does.
So p has _its own_ point q

A good way to show that,
if q₂ = q, then p₂ = p
is to show that
the p-to-q procedure is reversible
So each point q in ℚ⁺ has its own
point p in (a,b)

I have left the actual showing of
these things we should show
in favor of explaining why
what we show shows what we say it shows.

>>> You said that ℵ₀ points are passed
>>> before you can stop. Do they exist?
>>
>> Let q be an element of ℚ⁺
>
> No, the question was: Do they exist?

Suppose I describe some points.
I make some not-possibly-false-about-them claims.

If you can't bring yourself to say they exist,
according to however you mean "exist",
then I can't answer your question in a way
which you won't misunderstand.

On Saturday, January 28, 2023 at 7:06:13 AM UTC-4, WM wrote:
> William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> >
> > > Find a Peano element that has less [than ℵo] successors. Fail..
> >
> > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.

Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P. U_P has cardinality ℵo.

Jim Burns schrieb am Samstag, 28. Januar 2023 um 15:36:02 UTC+1:
> On 1/28/2023 6:11 AM, WM wrote:
> > Jim Burns schrieb am Samstag,
> > 28. Januar 2023 um 00:33:43 UTC+1:
> >> You're asking
> >> why is there no interval with
> >> finitely-many rationals in it?
> >
> > Yes.
>
> Because

Your explanation is batantly wrong. Proof: you cannot keep Bob and you cannot delete Bob. That is a contradiction. But it is irrelevant.

> >>> You said that ℵ₀ points are passed
> >>> before you can stop. Do they exist?
> >>
> >> Let q be an element of ℚ⁺
> >
> > No, the question was: Do they exist?
> Suppose I describe some points.

You cannot describe more than finitely many. Therefore don't try but mark the first wrong statement below:

1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
3) Its size differs according to the effort of the observer.
4) The difference can be reduced but contains almost all, namely ℵo unit fractions in any case.

Regards, WM

William schrieb am Samstag, 28. Januar 2023 um 16:07:12 UTC+1:
> On Saturday, January 28, 2023 at 7:06:13 AM UTC-4, WM wrote:
> > William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> > >
> > > > Find a Peano element that has less [than ℵo] successors. Fail.
> > >
> > > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> > 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> > 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
> Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.

Yes, that is true. But why do you say incorrect? Can you discern more U_P?

> U_P has cardinality ℵo.

Sínce you cannot define an x such that (x,1] contain infinitely many U_P, you cannot define most U_P.

Regards, WM

JVR schrieb am Samstag, 28. Januar 2023 um 12:46:33 UTC+1:
> On Saturday, January 28, 2023 at 12:36:05 PM UTC+1, WM wrote:

> > > > Would you be so kind to note which step is wrong in your opinion?
> > > 2, 3, and 4 are the usual meaningless garble.
> > > Define your terms.
> > Define a unit fraction such that the difference disappears. Then you will learn what defining means.
> >
> Almost nobody will notice.

You err. My PDF Dark numbers II in Research Gate has by far more than 2000 reads.

Regards, WM

On Saturday, January 28, 2023 at 1:50:45 PM UTC-4, WM wrote:
> William schrieb am Samstag, 28. Januar 2023 um 16:07:12 UTC+1:
> > On Saturday, January 28, 2023 at 7:06:13 AM UTC-4, WM wrote:
> > > William schrieb am Freitag, 27. Januar 2023 um 20:38:22 UTC+1:
> > > > On Friday, January 27, 2023 at 3:17:43 PM UTC-4, WM wrote:
> > > >
> > > > > Find a Peano element that has less [than ℵo] successors. Fail.
> > > >
> > > > Correct. In particular the first element has ℵo successors. Each is in the Peano set. The Peano set has Cardinality ℵo.
> > > 1) The half-open intervall (0, 1] is the minimal interval containing all unit fractions. (Nothing can be deleted at the edges.)
> > > 2) All unit fractions which can be discerned and individually noted sit in a smaller interval.
> > Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.
> Yes, that is true. But why do you say incorrect? Can you discern more U_P?

Yes, Every subset of U_P of cardinality 1 has the property that it can be written down (Note this does not imply that there is a subset of U_P of cardinality ℵo that has the property that it can be written down). U_P, a Peano set, has cardinality ℵo.

William schrieb am Samstag, 28. Januar 2023 um 20:19:32 UTC+1:
> On Saturday, January 28, 2023 at 1:50:45 PM UTC-4, WM wrote:

> > > Incorrect. For any real 0<x<1 the interval (x,1] contains only a finite number of elements of U_P.
> > Yes, that is true. But why do you say incorrect? Can you discern more U_P?
> Yes, Every subset of U_P of cardinality 1 has the property that it can be written down

No. All subsets of cardinality 1 which belong to the difference*) cannot be written down.

*) The half-open intervall (0, 1] is the minimal interval containing all unit fractions 1/n. (Nothing can be deleted at the edges.)
All unit fractions which can be discerned and individually noted or remembered belong to a smaller interval.
Its size differs according to the effort of the observer.
The difference can be reduced but contains almost all, namely ℵ0 unit fractions in any case.

Regards, WM

On Saturday, January 28, 2023 at 6:53:27 PM UTC+1, WM wrote:
> JVR schrieb am Samstag, 28. Januar 2023 um 12:46:33 UTC+1:
> > On Saturday, January 28, 2023 at 12:36:05 PM UTC+1, WM wrote:
>
> > > > > Would you be so kind to note which step is wrong in your opinion?
> > > > 2, 3, and 4 are the usual meaningless garble.
> > > > Define your terms.
> > > Define a unit fraction such that the difference disappears. Then you will learn what defining means.
> > >
> > Almost nobody will notice.
>
> You err. My PDF Dark numbers II in Research Gate has by far more than 2000 reads.
>
> Regards, WM

Yes, I hear that the AMS is about to fire all it's editors because they refused to publish your
fascinating research papers. The discovery of the dark numbers!! Now every bank and every
political party will want to hire you in order to run their invisible accounts.