William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
> On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
> > > On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
> > >
> > > > " We know for sure the "last unit faction" must be infinitesimal and cannot be an element of |R.
> > > > But we know that this is true for infinitely many unit fractions.
> > > This does not change the fact that "the last unit fraction" must be infinitesimal and cannot be an element of |R.
> > It cannot be an element of |R
> Correct. Also "the last unit fraction" must be infinitesimal.

That means not definable?
And what about the reciprocals?

Regards, WM

Jim Burns schrieb am Donnerstag, 2. Februar 2023 um 21:43:48 UTC+1:
> On 2/2/2023 11:46 AM, WM wrote:

Please limit your contributions to 50 lines. I do never read more.
>
> Therefore,
> no dark point is 1 or is between 0 and 1.

Then define the last point next to 0. You can't. But it is there if the real axis is continuous.
>
> When we consider claims about the elements of (0,1]
> no dark points are among _the ones we mean_

Then you are too superficial.

> There is no gap and there are no dark numbers
> in _what we mean_ by "line".
>
Then define any point of your choice and the point next to it.

Regards, WM

On Friday, February 3, 2023 at 6:24:46 AM UTC-4, WM wrote:
> William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
> > On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
> > > > On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
> > > > > William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
> > > >
> > > > > " We know for sure the "last unit faction" must be infinitesimal and cannot be an element of |R.
> > > > > But we know that this is true for infinitely many unit fractions.
> > > > This does not change the fact that "the last unit fraction" must be infinitesimal and cannot be an element of |R.
> > > It cannot be an element of |R
> > Correct. Also "the last unit fraction" must be infinitesimal.
> That means not definable?

Nope. Nothing to do with "can be written down". An infinitesimal, y, has the property that, given any real number x, y is between 0 and x. "The last unit faction" must have this property

On 2/3/2023 5:30 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 2. Februar 2023 um 21:43:48 UTC+1:

>> Therefore,
>> no dark point is 1 or is between 0 and 1.
>
> Then define the last point next to 0.
> You can't.

We know that
there is no last "visible" or "dark" point
next to 0.

We know there is no last point
because
we know some things about _what we mean_
by "line"
For example, what we mean by "line" only has
points which are finitely distinguishable.

Also,
we know that each claim in a sequence of
only not-first-possibly-false claims about
points in "line"
is a not-possibly-false claim about
points in "line".
So, we know those claims, too.

> But it is there if the real axis is continuous.

The real axis does not permit
continuous real-valued functions of reals
to jump over points.
(intermediate value theorem)

One consequence of having the not-jumping-over
property is that
for each split of the real axis,
there is a point between the before-part and
the after-part.

We know that these points are there
between the parts of splits
because it is another thing we know about
_what we mean_ by "line".

> But it is there if the real axis is continuous.

The real axis does not have
points which aren't finitely-indistinguishable
and does have
points between splits.

Things for which those aren't true
are something else, not the real axis.

>> When we consider claims about the elements of (0,1]
>> no dark points are among _the ones we mean_
>
> Then you are too superficial.

I am exactly as superficial as I need to be
in order to talk about _the ones we mean_

At a different time, in a different discussion,
a different level of superficiality would be needed
in order to discuss different _ones we mean_

What nearly-empty Pacific island have you (WM)
spent your entire life alone on, such that
that needs to be explained to you?

Consider finite ordinals and
not-necessarily-finite ordinals.

For a finite ordinal n
for each split between 0 and n
some i,j is last-before,first-after.

For a not-necessarily-finite ordinal n
for each split between 0 and n
some ⬚j is ⬚⬚⬚⬚⬚⬚⬚⬚⬚first-after.

For finite and not-necessarily-finite
ordinals i,j,
i < j and no other ordinal between them,
write j = i+1

Imagine discussing finite ordinals.
| n is a finite ordinal.

I am exactly as superficial as I need to be
in order to claim not-possibly-falsely
that, for each split between 0 and n
some i is last-before.

I can augment a not-possibly-false claim
about n with visibly not-first-possibly-false
claims about n, and each of those claims
are not-possibly-false about each one of
infinitely-many finite ordinals.

Augmenting with visibly not-first-possibly-false
is how we know _without checking each one_ that
no finite ordinal is the numerator of sqrt(2)
It's how we know sqrt(2) is irrational.

If we made that claim "less superficially"
about more-than-all finite ordinals, it would
no longer a not-possibly-false, and we couldn't
use it in the same way, and we wouldn't know that
sqrt(2) is irrational.

So, I am exactly as superficial as I need to be.

On 2/3/2023 5:30 AM, WM wrote:

> Please limit your contributions to 50 lines.
> I do never read more.

Do better, Wolfgang Mückenheim of
Hochschule Augsburg.

It happens that William formulated :
> On Friday, February 3, 2023 at 6:24:46 AM UTC-4, WM wrote:
>> William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
>>> On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
>>>> William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
>>>>> On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
>>>>>> William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
>>>>>> " We know for sure the "last unit faction" must be infinitesimal and
>>>>>> cannot be an element of |R. But we know that this is true for
>>>>>> infinitely many unit fractions.
>>>>> This does not change the fact that "the last unit fraction" must be
>>>>> infinitesimal and cannot be an element of |R.
>>>> It cannot be an element of |R
>>> Correct. Also "the last unit fraction" must be infinitesimal.
>> That means not definable?
>
> Nope. Nothing to do with "can be written down". An infinitesimal, y, has
> the property that, given any real number x, y is between 0 and x. "The last
> unit faction" must have this property

https://en.wikipedia.org/wiki/Hyperreal_number

There are no infinitesimals in the reals.

On 2/3/2023 12:18 PM, FromTheRafters wrote:
> It happens that William formulated :
>> On Friday, February 3, 2023
>> at 6:24:46 AM UTC-4, WM wrote:

>>> That means not definable?
>>
>> Nope.
>> Nothing to do with "can be written down".
>> An infinitesimal, y, has the property that,
>> given any real number x, y is between 0 and x.
>> "The last unit faction" must have this property
>
> https://en.wikipedia.org/wiki/Hyperreal_number
>
> There are no infinitesimals in the reals.

That does not deter WM.

WM wants
| x is a real number
to mean
| x is a real number or something else.

He considers the analysis of real numbers as
real numbers "too superficial".

Message-ID: <3e749282-c34f-4375-8fbf-9d6231910ab9n@googlegroups.com>
Subject: Re: The testimony of unit fractions
Date: Fri, 3 Feb 2023 02:30:51 -0800 (PST)

William schrieb am Freitag, 3. Februar 2023 um 15:31:11 UTC+1:
> On Friday, February 3, 2023 at 6:24:46 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
> > > On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
> > > > > On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
> > > > > > William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
> > > > >
> > > > > > " We know for sure the "last unit faction" must be infinitesimal and cannot be an element of |R.
> > > > > > But we know that this is true for infinitely many unit fractions.
> > > > > This does not change the fact that "the last unit fraction" must be infinitesimal and cannot be an element of |R.
> > > > It cannot be an element of |R
> > > Correct. Also "the last unit fraction" must be infinitesimal.
> > That means not definable?
> Nope. Nothing to do with "can be written down". An infinitesimal, y, has the property that, given any real number x, y is between 0 and x. "The last unit faction" must have this property

And infinitely many of its colleagues must have this property too. Can you distingish them? And what about their reciprocals?

Regards, WM

Jim Burns schrieb am Freitag, 3. Februar 2023 um 16:04:46 UTC+1:
> On 2/3/2023 5:30 AM, WM wrote:
> > Jim Burns schrieb am Donnerstag,
> > 2. Februar 2023 um 21:43:48 UTC+1:
> >> Therefore,
> >> no dark point is 1 or is between 0 and 1.
> >
> > Then define the last point next to 0.
> > You can't.
> We know that
> there is no last "visible" or "dark" point
> next to 0.

No, you believe that since you could not determine it. But if the real line is continuos, then it must be there.
>
> We know there is no last point
> because
> we know some things about _what we mean_
> by "line"
> For example, what we mean by "line" only has
> points which are finitely distinguishable.

Next to zero there is either a point or nothing.
>
> > But it is there if the real axis is continuous.
> The real axis does not permit
> continuous real-valued functions of reals
> to jump over points.
> (intermediate value theorem)

If there is nothing next to zero, then the intermediate value theorem is wrong.

Regards, WM

On Friday, February 3, 2023 at 3:26:49 PM UTC-4, WM wrote:
> William schrieb am Freitag, 3. Februar 2023 um 15:31:11 UTC+1:
> > On Friday, February 3, 2023 at 6:24:46 AM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
> > > > On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
> > > > > William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
> > > > > > On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
> > > > > > > William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
> > > > > >
> > > > > > > " We know for sure the "last unit faction" must be infinitesimal and cannot be an element of |R.
> > > > > > > But we know that this is true for infinitely many unit fractions.
> > > > > > This does not change the fact that "the last unit fraction" must be infinitesimal and cannot be an element of |R.
> > > > > It cannot be an element of |R
> > > > Correct. Also "the last unit fraction" must be infinitesimal.
> > > That means not definable?
> > Nope. Nothing to do with "can be written down". An infinitesimal, y, has the property that, given any real number x, y is between 0 and x. "The last unit faction" must have this property
> And infinitely many of its colleagues must have this property too.

Note the "too" , which means of course you agree that "the first unit fraction" must be infinitesimal, and not an element of |R.

William schrieb am Freitag, 3. Februar 2023 um 20:59:13 UTC+1:
> On Friday, February 3, 2023 at 3:26:49 PM UTC-4, WM wrote:
> > William schrieb am Freitag, 3. Februar 2023 um 15:31:11 UTC+1:
> > > On Friday, February 3, 2023 at 6:24:46 AM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 2. Februar 2023 um 19:27:19 UTC+1:
> > > > > On Thursday, February 2, 2023 at 2:18:03 PM UTC-4, WM wrote:
> > > > > > William schrieb am Donnerstag, 2. Februar 2023 um 17:53:57 UTC+1:
> > > > > > > On Thursday, February 2, 2023 at 12:36:21 PM UTC-4, WM wrote:
> > > > > > > > William schrieb am Donnerstag, 2. Februar 2023 um 16:00:16 UTC+1:
> > > > > > >
> > > > > > > > " We know for sure the "last unit faction" must be infinitesimal and cannot be an element of |R.
> > > > > > > > But we know that this is true for infinitely many unit fractions.
> > > > > > > This does not change the fact that "the last unit fraction" must be infinitesimal and cannot be an element of |R.
> > > > > > It cannot be an element of |R
> > > > > Correct. Also "the last unit fraction" must be infinitesimal.
> > > > That means not definable?
> > > Nope. Nothing to do with "can be written down". An infinitesimal, y, has the property that, given any real number x, y is between 0 and x. "The last unit faction" must have this property
> > And infinitely many of its colleagues must have this property too.
> Note the "too" , which means of course you agree that "the first unit fraction" must be infinitesimal, and not an element of |R.

Of course I agree that it is not a real number in the common sense of the word, although it is a point lying on the real axis. How we call it is less important. The name should however express that it is of different kind.

Regards, WM

On Friday, February 3, 2023 at 5:51:31 PM UTC-4, WM wrote:

> Of course I agree that it is not a real number

Indeed. A "dark real number" is not a real number. A "first unit fraction" must be infinitesimal.

William schrieb am Freitag, 3. Februar 2023 um 23:25:33 UTC+1:
> On Friday, February 3, 2023 at 5:51:31 PM UTC-4, WM wrote:
>
> > Of course I agree that it is not a real number
> Indeed. A "dark real number" is not a real number. A "first unit fraction" must be infinitesimal.

Well, this is an interesting topic. Are you interested to discuss the consequences?
Are there ℵo infinitesimal unit fractions?
What about their reciprocals?
What about their points on the real axis?

Regards, WM

On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> William schrieb am Freitag, 3. Februar 2023 um 23:25:33 UTC+1:
> > On Friday, February 3, 2023 at 5:51:31 PM UTC-4, WM wrote:
> >
> > > Of course I agree that it is not a real number
> > Indeed. A "dark real number" is not a real number. A "first unit fraction" must be infinitesimal.
> Well, this is an interesting topic. Are you interested to discuss the consequences?

A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.

> Are there ℵo infinitesimal unit fractions?

Indeed, and not one "infinitesimal unit fraction" is a unit fraction.

> What about their reciprocals?

They are "dark natural numbers". Not one of them is a natural number.

On 2/3/2023 2:30 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 3. Februar 2023 um 16:04:46 UTC+1:
>> On 2/3/2023 5:30 AM, WM wrote:

>>> But it is there if the real axis is continuous.
>>
>> The real axis does not permit
>> continuous real-valued functions of reals
>> to jump over points.
>> (intermediate value theorem)
>
> If there is nothing next to zero,
> then the intermediate value theorem is wrong.

No,
if there is anything next to zero,
then the intermediate value theorem is wrong.

----

_if_
| | some d is next to 0 in "line"
| some positive point d exists
with nothing between 0 and d
_then_
| | "line" hasn't the Archimedean property
| some positive d < each finite 1/n

_if_
| | "line" hasn't the Archimedean property
| some positive d < each finite 1/n
_then_
| | "line" hasn't the LUB property
| some bounded non-∅ set S of reals
does not have a least-upper-bound
[1]

_if_
| | "line" hasn't the LUB property
| some bounded non-∅ set S of reals
does not have a least-upper-bound
_then_
| | intermediate value theorem is wrong for "line"
| a function exists which is continuous everywhere
and which also jumps over points,
[2]

----
[1]

| Assume some positive d < each finite 1/n
| | D is the set of all positive d < each finite 1/n
| each such d < 1
| D is a bounded non-∅ set S of reals
| | No b = LUB of D exists
| || Assume otherwise.
|| Assume b = LUB of D
||
|| 0 < b/2 < b < 2b
||
|| b/2 < each finite 1/n
|| 2b > some finite 1/n
||
|| However,
|| either
|| b/2 b 2b all < each finite 1/n
|| or
|| b/2 b 2b all > some finite 1/n
|| Contradiction.
| | Therefore,
| "line" hasn't the LUB property

Therefore,
if "line" hasn't the Archimedean property
then "line" hasn't the LUB property

----
[2]

| Assume a bounded non-empty set D of reals
| exists
| for which no least-upper-bound exists.
| | Define f: R -> {0,1}
| as
| f(x) = 1 if x is a bound of D
| f(x) = 0 if x is not a bound of D
| | f is continuous everywhere except at
| some hypothetical point b with bounds on
| one side and non-bounds on the other side.
| Such a point b would be LUB of D
| | However,
| LUB of D doesn't exist.
| Thus, f is continuous everywhere
| and jumps over 1/2

Therefore,
if "line" hasn't the LUB property
then intermediate value theorem is wrong
for "line"

Jim Burns schrieb am Samstag, 4. Februar 2023 um 16:07:34 UTC+1:
> On 2/3/2023 2:30 PM, WM wrote:

> > If there is nothing next to zero,
> > then the intermediate value theorem is wrong.
> No,
> if there is anything next to zero,
> then the intermediate value theorem is wrong.

There is nothing next to zero that could be distinguished from points next to itself.
The Archimedean property holds only for definable points.
> ----
>
> _if_
> |
> | some d is next to 0 in "line"
> |
> some positive point d exists
> with nothing between 0 and d

This point exists but cannot be distinguished from infinitely many points in the immediate vicinity of 0.

Regards, WM

William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:

> > Well, this is an interesting topic. Are you interested to discuss the consequences?
> A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.

But between all real unit fractions and zero there are dark unit fractions.

> > Are there ℵo infinitesimal unit fractions?
> Indeed, and not one "infinitesimal unit fraction" is a unit fraction.

The infinitesimal unit fractions are residing between the real unit fractions and zero.
>
> > What about their reciprocals?
>
> They are "dark natural numbers". Not one of them is a natural number.

They are residing between the Peano numbers and ω.

Regards, WM

On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
>
> > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> But between all real unit fractions and zero there are dark unit fractions.
So what? "Dark unit fractions" are still not unit fractions.
> > > Are there ℵo infinitesimal unit fractions?
> > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> The infinitesimal unit fractions are residing between the real unit fractions and zero.
> >
> > > What about their reciprocals?
> >
> > They are "dark natural numbers". Not one of them is a natural number.
> They are residing between the Peano numbers and ω.
So what? They, "dark natural numbers", are still not natural numbers.

William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
> On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> >
> > > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> > But between all real unit fractions and zero there are dark unit fractions.
> So what? "Dark unit fractions" are still not unit fractions.
> > > > Are there ℵo infinitesimal unit fractions?
> > > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> > The infinitesimal unit fractions are residing between the real unit fractions and zero.
> > >
> > > > What about their reciprocals?
> > >
> > > They are "dark natural numbers". Not one of them is a natural number.
> > They are residing between the Peano numbers and ω.
> So what? They, "dark natural numbers", are still not natural numbers.

I agree that they are not Peano-natural-numbers. But they are there. Why not investigate them?

Regards, WM

On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
> William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
> > On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > > > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> > >
> > > > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > > > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> > > But between all real unit fractions and zero there are dark unit fractions.
> > So what? "Dark unit fractions" are still not unit fractions.
> > > > > Are there ℵo infinitesimal unit fractions?
> > > > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> > > The infinitesimal unit fractions are residing between the real unit fractions and zero.
> > > >
> > > > > What about their reciprocals?
> > > >
> > > > They are "dark natural numbers". Not one of them is a natural number.
> > > They are residing between the Peano numbers and ω.
> > So what? They, "dark natural numbers", are still not natural numbers.
> I agree that they are not Peano-natural-numbers.

They are not any type of natural numbers. "Dark natural numbers" are not natural numbers.

On 2/4/2023 12:59 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 4. Februar 2023 um 16:07:34 UTC+1:

>> _if_
>> |
>> | some d is next to 0 in "line"
>> |
>> some positive point d exists
>> with nothing between 0 and d
>
> This point exists

Then,
what you are talking about is
d with nothing between it and 0
which is d < each finite 1/n
and a bounded non-∅ set D with no LUB
and a continuous f(x) which jumps over 1/2

What we are talking about are
continuous functions which don't jump.

You (WM) have expressed concern that we, by
talking about what we're talking about,
instead of what you're talking about,
are "too superficial".

We are precisely as superficial as we need to be
in order to make not-possibly-false claims
about "line" such that continuous functions
do not jump.

By augmenting those not-possibly-false claims
about "line" with visibly not-first-possibly-false
claims about "line",

we can discover and _know_ fresh-to-us claims
about "line",
without checking infinitely-many points.

> This point exists
> but cannot be distinguished from
> infinitely many points in
> the immediate vicinity of 0.

What we are talking about are
continuous functions which don't jump.

William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
> On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
> > > On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> > > > William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > > > > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> > > >
> > > > > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > > > > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> > > > But between all real unit fractions and zero there are dark unit fractions.
> > > So what? "Dark unit fractions" are still not unit fractions.
> > > > > > Are there ℵo infinitesimal unit fractions?
> > > > > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> > > > The infinitesimal unit fractions are residing between the real unit fractions and zero.
> > > > >
> > > > > > What about their reciprocals?
> > > > >
> > > > > They are "dark natural numbers". Not one of them is a natural number.
> > > > They are residing between the Peano numbers and ω.
> > > So what? They, "dark natural numbers", are still not natural numbers.
> > I agree that they are not Peano-natural-numbers.
> They are not any type of natural numbers. "Dark natural numbers" are not natural numbers.

They are, as you call them, dark natural numbers. Don't be a racist. You cannot exclude the dark unit fractions from the real axis.

Regards, WM

Jim Burns schrieb am Samstag, 4. Februar 2023 um 20:42:01 UTC+1:

> what you are talking about is
> d with nothing between it and 0
> which is d < each finite 1/n

No problem. Nobody can use it and its infinitely many colleagues.
We only know from set theory that there are all unit fractions and from the cursor that the complete set of unit fractions cannot be recognized.

Regards, WM

On Saturday, February 4, 2023 at 6:37:31 PM UTC-4, WM wrote:
> William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
> > On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
> > > > On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> > > > > William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > > > > > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> > > > >
> > > > > > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > > > > > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> > > > > But between all real unit fractions and zero there are dark unit fractions.
> > > > So what? "Dark unit fractions" are still not unit fractions.
> > > > > > > Are there ℵo infinitesimal unit fractions?
> > > > > > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> > > > > The infinitesimal unit fractions are residing between the real unit fractions and zero.
> > > > > >
> > > > > > > What about their reciprocals?
> > > > > >
> > > > > > They are "dark natural numbers". Not one of them is a natural number.
> > > > > They are residing between the Peano numbers and ω.
> > > > So what? They, "dark natural numbers", are still not natural numbers.
> > > I agree that they are not Peano-natural-numbers.
> > They are not any type of natural numbers. "Dark natural numbers" are not natural numbers.
> They are, as you call them, dark natural numbers. Don't be a racist. You cannot exclude the dark unit fractions from the real axis.
>

The real number line contains only points corresponding to elements of |R. It does not contain any points corresponding to "dark natural numbers" which are not natural numbers. Nor does it contain any points corresponding to "infinitesimal unit fractions"