Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.

For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to cover it, we have a convincing proof of dark unit fractions in the vicinity of zero.

The set has NUF 0. But this can only be checked by considering its elements because the point set is nothing more than its elements. For every considered point is fails. Hence not all points can be considered.

A kinetic point of view: Move the cursor from 1 to 0 and note all unit fractions. When arriving at 0 you have passed ℵ₀ unit fractions in linear order. If they all were well-ordered, then you would have passed a last one (because no more are following). But you cannot determine which was the last one. That proves that most unit fractions are dark, i.e., not individually accessible.

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.

Regards, WM

WM presented the following explanation :
> Let us denote the Number of Unit Fractions existing between an object O on
> the positive real axis and zero as NUF_O.
>
> For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
>
> Since all unit fractions cover the interval (0, 1] but the definable unit
> fractions fail to cover it, we have a convincing proof of dark unit fractions
> in the vicinity of zero.
>
> The set has NUF 0. But this can only be checked by considering its elements
> because the point set is nothing more than its elements. For every considered
> point is fails. Hence not all points can be considered.

Are cluster points necessarily unit fractions? Are you guaranteed
convergence in Q?

On Friday, January 27, 2023 at 11:10:04 AM UTC+1, WM wrote:
> Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
>
> For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
>
> Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to cover it, we have a convincing proof of dark unit fractions in the vicinity of zero.
>
> The set has NUF 0. But this can only be checked by considering its elements because the point set is nothing more than its elements. For every considered point is fails. Hence not all points can be considered.
>
> A kinetic point of view: Move the cursor from 1 to 0 and note all unit fractions. When arriving at 0 you have passed ℵ₀ unit fractions in linear order. If they all were well-ordered, then you would have passed a last one (because no more are following). But you cannot determine which was the last one. That proves that most unit fractions are dark, i.e., not individually accessible.
>
> 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.
>
> Regards, WM

It's astonishing to what extent quackery in general, and Mückmeatical quackery in particular, depends upon neologisms and
undefined terms, as well is the misuse of the common vocabulary. For example, in the murky nonsense that Mücke posted above,
almost every phrase is murky:

- number of (when more than finitely many)
- to be checked
- by considering
- considered point
- it fails
- an object O
- real axis (when neither naturals nor rationals exist in the usual sense)
- we find
- ℵo
- definable
- cover
- fail to cover
- dark
- dark unit fractions
- vicinity

some declarative sentences:
- The set has NUF 0
- can only be checked by considering its elements
- for every considered point it fails
- hence not all points can be considered

- cursor
- move
- note
- arriving at
- pass
- linear order
- well ordered
- last one
- no more are following
- you cannot determine
- that proves
- most unit fractions are dark
- individually accessible
- follow
- last one passed
- last one noted
- hence
- has a finite number
etc etc

That proves it: You cannot nail a pudding to the wall. But you can make a salad out of words.

FromTheRafters schrieb am Freitag, 27. Januar 2023 um 13:31:45 UTC+1:
> WM presented the following explanation :
> > Let us denote the Number of Unit Fractions existing between an object O on
> > the positive real axis and zero as NUF_O.
> >
> > For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> > For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> > But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
> >
> > Since all unit fractions cover the interval (0, 1] but the definable unit
> > fractions fail to cover it, we have a convincing proof of dark unit fractions
> > in the vicinity of zero.
> >
> > The set has NUF 0. But this can only be checked by considering its elements
> > because the point set is nothing more than its elements. For every considered
> > point is fails. Hence not all points can be considered.
> Are cluster points necessarily unit fractions? Are you guaranteed
> convergence in Q?

What do you call a cluster point?
Why is convergence important?

When the cursor moves from 1 to 0, its velocity can be arbitrary and can be varied arbitrarily. The only condition is that it starts and arrives within an arbitrarily chosen finite time interval.

Regards, WM

JVR schrieb am Freitag, 27. Januar 2023 um 14:06:49 UTC+1:

> - number of (when more than finitely many)

erhalten wir nicht bloß eine einzige unendliche ganze Zahl, sondern eine unendliche Folge von solchen, die voneinander wohl unterschieden sind [Cantor]
You have never read Cantor? Or only never understood what you read, like my text here?

> - to be checked
> - by considering
> - considered point
> - it fails
> - an object O
> - real axis (when neither naturals nor rationals exist in the usual sense)
> - we find
> - ℵo
> - definable
> - cover
> - fail to cover
> - dark
> - dark unit fractions
> - vicinity
>
> some declarative sentences:
> - The set has NUF 0
> - can only be checked by considering its elements
> - for every considered point it fails
> - hence not all points can be considered
>
> - cursor
> - move
> - note
> - arriving at
> - pass
> - linear order
> - well ordered
> - last one
> - no more are following
> - you cannot determine
> - that proves
> - most unit fractions are dark
> - individually accessible
> - follow
> - last one passed
> - last one noted
> - hence
> - has a finite number
> etc etc
>
> That proves it: You cannot nail a pudding to the wall. But you can make a salad out of words.

Bravo! Sure, you are a great salad master.
https://www.123rf.com/photo_131089519_chef-in-the-kitchen-of-the-hotel-or-restaurant-decorates-the-food-just-before-serving-.html?vti=mxxc23zbi9hrufqdgn-1-15

Regards, WM

https://www.youtube.com/watch?v=VjFyX2sSTGs

On Friday, 27 January 2023 at 06:10:04 UTC-4, WM wrote:
> Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.

You can't do that!!! First you have to define a distance between points on the real axis, and THEN you can figure out how to apply this to distances of sets. Hint: For every point in (0,oo) the distance from 0 in your metric is ℵo. The only sensible way tog et a distance of a set is to use the infimum. Unfortunately for you, inf{x in (0,oo)} d(x,0) = ℵo.

But, of course you don't give a shit about quantifier shifts. So please fuck off.

> For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
>
> Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to cover it, we have a convincing proof of dark unit fractions in the vicinity of zero.
>
> The set has NUF 0. But this can only be checked by considering its elements because the point set is nothing more than its elements. For every considered point is fails. Hence not all points can be considered.
>
> A kinetic point of view: Move the cursor from 1 to 0 and note all unit fractions. When arriving at 0 you have passed ℵ₀ unit fractions in linear order. If they all were well-ordered, then you would have passed a last one (because no more are following). But you cannot determine which was the last one. That proves that most unit fractions are dark, i.e., not individually accessible.
>
> 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.
>
> Regards, WM

WM explained on 1/27/2023 :
> FromTheRafters schrieb am Freitag, 27. Januar 2023 um 13:31:45 UTC+1:
>> WM presented the following explanation :
>>> Let us denote the Number of Unit Fractions existing between an object O on
>>> the positive real axis and zero as NUF_O.
>>>
>>> For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
>>> For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
>>> But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
>>>
>>> Since all unit fractions cover the interval (0, 1] but the definable unit
>>> fractions fail to cover it, we have a convincing proof of dark unit
>>> fractions in the vicinity of zero.
>>>
>>> The set has NUF 0. But this can only be checked by considering its elements
>>> because the point set is nothing more than its elements. For every
>>> considered point is fails. Hence not all points can be considered.
>> Are cluster points necessarily unit fractions? Are you guaranteed
>> convergence in Q?
>
> What do you call a cluster point?

Points in the neighborhood of zero which aren't isolated.

> Why is convergence important?

How can any regular sequence of non-zero rational values 'get to zero'
in a fractional step-by-step process when zero itself is not in the
set? Many such sequences in the rationals will fail to converge because
the rationals are not complete.

> When the cursor moves from 1 to 0, its velocity can be arbitrary and can be
> varied arbitrarily. The only condition is that it starts and arrives within
> an arbitrarily chosen finite time interval.

The powerful Axiom of Because WM Said So.

On 1/27/2023 9:23 AM, FredJeffries wrote:
> https://www.youtube.com/watch?v=VjFyX2sSTGs

Has to be the booze wrt WM? Humm... Or his he totally sober? A true idiot?

Gus Gassmann schrieb am Freitag, 27. Januar 2023 um 20:21:39 UTC+1:
> On Friday, 27 January 2023 at 06:10:04 UTC-4, WM wrote:
> > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> You can't do that!!!

Of course I can and I did:
For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

> First you have to define a distance between points on the real axis,

That metric (called real metric) should be known by every reader here. If not, see my book W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015).
> Hint: For every point in (0,oo) the distance from 0 in your metric is ℵo.

No. The interval (0, 1] is the smallest Interval containing all points of unit fractions. Hence nothing can be removed. Not any single point.

> The only sensible way tog et a distance of a set is to use the infimum.

The real metric shows that there is no point between 0 and (0, 1]. Hence there cannot exist a unit fraction between 0 and (0, 1]. Hence the NUF_(o, 1] is 0 too.

Unfortunately for you, inf{x in (0,oo)} d(x,0) = ℵo.

Unfortunately for you that is circumvented by the real metric.
>
> But, of course you don't give a shit about quantifier shifts.

I do not use it. I use mathematics and observe geometry:

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

FromTheRafters schrieb am Freitag, 27. Januar 2023 um 20:50:08 UTC+1:
> WM explained on 1/27/2023 :

> >> Are cluster points necessarily unit fractions? Are you guaranteed
> >> convergence in Q?
> >
> > What do you call a cluster point?
> Points in the neighborhood of zero which aren't isolated.

So you accept points which cannot be isolated? That is waht I prove:

> How can any regular sequence of non-zero rational values 'get to zero'
> in a fractional step-by-step process when zero itself is not in the
> set? Many such sequences in the rationals will fail to converge because
> the rationals are not complete.

The sequence of unit fractions converges. It is complete according to set theory. We get:
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.

> The powerful Axiom of Because WM Said So.

What step of the above usese this axiom? I have enumerated them for easier communication.

Regards, WM

WM formulated the question :
> FromTheRafters schrieb am Freitag, 27. Januar 2023 um 20:50:08 UTC+1:
>> WM explained on 1/27/2023 :
>
>>>> Are cluster points necessarily unit fractions? Are you guaranteed
>>>> convergence in Q?
>>>
>>> What do you call a cluster point?
>> Points in the neighborhood of zero which aren't isolated.
>
> So you accept points which cannot be isolated? That is waht I prove:
>
>> How can any regular sequence of non-zero rational values 'get to zero'
>> in a fractional step-by-step process when zero itself is not in the
>> set? Many such sequences in the rationals will fail to converge because
>> the rationals are not complete.
>
> The sequence of unit fractions converges.

No, it doesn't. It would in the reals though unless you restricted your
domain of discourse to the positive reals. Your sequence converges to a
point outside the set, so we say it diverges.

> It is complete according to set theory.

Again, you don't seem to know what 'complete' means in this context.

https://math.stackexchange.com/questions/2259847/confused-about-incompleteness-of-rational-numbers

On Saturday, January 28, 2023 at 1:14:30 PM UTC+1, FromTheRafters wrote:
> WM formulated the question :
> > FromTheRafters schrieb am Freitag, 27. Januar 2023 um 20:50:08 UTC+1:
> >> WM explained on 1/27/2023 :
> >
> >>>> Are cluster points necessarily unit fractions? Are you guaranteed
> >>>> convergence in Q?
> >>>
> >>> What do you call a cluster point?
> >> Points in the neighborhood of zero which aren't isolated.
> >
> > So you accept points which cannot be isolated? That is waht I prove:
> >
> >> How can any regular sequence of non-zero rational values 'get to zero'
> >> in a fractional step-by-step process when zero itself is not in the
> >> set? Many such sequences in the rationals will fail to converge because
> >> the rationals are not complete.
> >
> > The sequence of unit fractions converges.
> No, it doesn't. It would in the reals though unless you restricted your
> domain of discourse to the positive reals. Your sequence converges to a
> point outside the set, so we say it diverges.
> > It is complete according to set theory.
> Again, you don't seem to know what 'complete' means in this context.
>
> https://math.stackexchange.com/questions/2259847/confused-about-incompleteness-of-rational-numbers

He's been told many times that 'complete' has a technical meaning in this context.
Mücke isn't here to learn. He is here to get his daily dose of abuse.

On Saturday, 28 January 2023 at 06:44:30 UTC-4, WM wrote:
> Gus Gassmann schrieb am Freitag, 27. Januar 2023 um 20:21:39 UTC+1:
> > On Friday, 27 January 2023 at 06:10:04 UTC-4, WM wrote:
> > > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> > You can't do that!!!
> Of course I can and I did

You cannot, and repeating your bullshit doesn't take away any of the stench.. You are a fucking moron, and have been for the best part of twenty years. You have zero understanding of proofs, you wouldn't recognize one if two of them bit you on the balls simultaneously, and you can't even copy one correctly.

And now FUCK OFF.

EOD

FromTheRafters schrieb am Samstag, 28. Januar 2023 um 13:14:30 UTC+1:
> WM formulated the question :
> > FromTheRafters schrieb am Freitag, 27. Januar 2023 um 20:50:08 UTC+1:
> >> WM explained on 1/27/2023 :
> >
> >>>> Are cluster points necessarily unit fractions? Are you guaranteed
> >>>> convergence in Q?
> >>>
> >>> What do you call a cluster point?
> >> Points in the neighborhood of zero which aren't isolated.
> >
> > So you accept points which cannot be isolated? That is waht I prove:
> >
> >> How can any regular sequence of non-zero rational values 'get to zero'
> >> in a fractional step-by-step process when zero itself is not in the
> >> set? Many such sequences in the rationals will fail to converge because
> >> the rationals are not complete.
> >
> > The sequence of unit fractions converges.
> No, it doesn't. It would in the reals though unless you restricted your
> domain of discourse to the positive reals.

The sequence converges in the reals to zero.

> Your sequence converges to a
> point outside the set, so we say it diverges.

Say what you like, it does not matter here.

> > It is complete according to set theory.
> Again, you don't seem to know what 'complete' means in this context.

You are confusing the notions. Complete means that all unit fractions are existing. It has nothing to do with:
>
> https://math.stackexchange.com/questions/2259847/confused-about-incompleteness-of-rational-numbers

In this respect the unit fractions are by far more incomplete than all fractions. But that is irrelevant.

Important is that all unit fractions exist in (0, 1] and that all unit fractions which can be discerned and individually noted sit in a smaller interval. Convergence is irrelevant.

Regards, WM

Gus Gassmann schrieb am Samstag, 28. Januar 2023 um 16:01:49 UTC+1:
> On Saturday, 28 January 2023 at 06:44:30 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Freitag, 27. Januar 2023 um 20:21:39 UTC+1:
> > > On Friday, 27 January 2023 at 06:10:04 UTC-4, WM wrote:
> > > > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> > > You can't do that!!!
> > Of course I can and I did
> You cannot,

I did. Why did you delete the examples?

For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

Do you have the illusion that you are right when you delete the counter evidence?

Regards, WM

JVR schrieb am Samstag, 28. Januar 2023 um 14:25:10 UTC+1:
> On Saturday, January 28, 2023 at 1:14:30 PM UTC+1, FromTheRafters wrote:

> > > It is complete according to set theory.
> > Again, you don't seem to know what 'complete' means in this context.

This context is irelevant for my proof.

> He's been told many times that 'complete' has a technical meaning in this context.

But "this context" does not matter for my proof. I only need that all unit fractions are existing in the half-open interval.

Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.

For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to crover it, we have a convincing proof of dark unit fractions in the vicinity of zero.

Regards, WM

On 1/28/2023 5:52 AM, WM wrote:
> FromTheRafters schrieb am Freitag,
> 27. Januar 2023 um 20:50:08 UTC+1:
>> WM explained on 1/27/2023 :

>>> What do you call a cluster point?
>>
>> Points in the neighborhood of zero which
>> aren't isolated.
>
> So you accept points which cannot be isolated?
> That is waht I prove:

For a set U,
we can designate certain subsets of U
to be _open_

Which sets we designate to be open fixes which
points in U we consider near each other.

A particular choice Openₖ(U) of which sets are open
is called a _topology_ of U

There are a few rules which Openₖ(U) must obey
in order to be a topology.
| U and ∅ are in Openₖ(U)
| Any union of open sets (sets in Openₖ(U))
| is an open set (is in Openₖ(U))
| The intersection of finitely-many open sets
| is an open set.

Different topologies Openⱼ(U) ≠ Openₖ(U) of U
claim different points of U are near each other.

> So you accept points which cannot be isolated?
> That is waht I prove:

A _cluster point_ x of U has,
in each _neighborhood_ N(x) of x,
some other point y ≠ x

A _neighborhood_ of x is an open set containing x.
Open sets say which points are near each other.
Thus,
x is a cluster point of U iff
other points of U are near x

An _isolated_ point is
the opposite of a cluster point.
x is an isolated point of U iff
not all neighborhoods (open sets) of x
contain some y ≠ x

x is isolated ⟺ {x} ∈ Openₖ(U)
x is a cluster point ⟺ {x} ∉ Openₖ(U)

----
This language of open sets, of neighborhoods,
of being near in U (according to Openₖ(U))
is good for expressing the idea of limit
without a lot of the baggage that comes with
the real numbers, baggage which you dispute.

Let ⟨ x₁ x₂ x₃ ... ⟩ be an infinite sequence of
points in U
There is a first point. There is no last point.
For each split of ⟨ x₁ x₂ x₃ ... ⟩
some xᵢ,xᵢ₊₁ is last-before,first-after.

| x is the _limit_ of ⟨ x₁ x₂ x₃ ... ⟩
| iff
| _almost all_ of ⟨ x₁ x₂ x₃ ... ⟩ is _near_ x

In the language of open sets, of neighborhoods,
| x is the _limit_ of ⟨ x₁ x₂ x₃ ... ⟩
| iff,
| for each neighborhood N(x) of x,
| if there is a first point y missed by N(x)
| then there is a last point z missed by N(x)
| == no more than finitely-many missed points.

> The sequence of unit fractions converges.

Yes.
That means that
almost all unit fractions in the sequence
are near 0

For each unit fraction 1/n in the sequence,
for each split between 1/1 and 1/n
some 1/i,1/i⁺⁺ is last-before,first-after.

Almost all unit fractions such that,
for each split between 1/1 and 1/n
some 1/i,1/i⁺⁺ is last-before,first-after
are near 0

> 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.

All unit fractions in the sequence,
which means
all unit fractions such that,
for each split between 1/1 and 1/n
some 1/i,1/i⁺⁺ is last-before,first-after,
sit in an interval which ends
_no non-zero distance_ from 0

If it were otherwise,
that would contradict the fact that
almost all unit fractions in the sequence
are near 0

On Friday, January 27, 2023 at 6:10:04 AM UTC-4, WM wrote:
> Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
>
> For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

This is of course true since
for every unit faction 1/n we find NUF_1/n= ℵo.

No need to restrict things to "definable"

William schrieb am Sonntag, 29. Januar 2023 um 00:48:37 UTC+1:
> On Friday, January 27, 2023 at 6:10:04 AM UTC-4, WM wrote:
> > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> >
> > For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
> > For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
> > But for every definable unit fraction 1/n we find NUF_1/n = ℵo.
> This is of course true since
> for every unit faction 1/n we find NUF_1/n= ℵo.

These ℵo unit fractions cannot not be resolved into individuals because ℵo will forever remain as a crowd.
>
> No need to restrict things to "definable"

Definable unif fractions can be separated from the crowd.

Regards, WM

Jim Burns schrieb am Samstag, 28. Januar 2023 um 23:22:26 UTC+1:
> On 1/28/2023 5:52 AM, WM wrote:

> > The sequence of unit fractions converges.
> Yes.
> That means that
> almost all unit fractions in the sequence
> are near 0

So near that they cannot be resolved into individuals.

> > 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.
> All unit fractions in the sequence,
> which means
> all unit fractions such that,
> for each split between 1/1 and 1/n
> some 1/i,1/i⁺⁺ is last-before,first-after,
> sit in an interval which ends
> _no non-zero distance_ from 0

The difference is dark.
>
> If it were otherwise,
> that would contradict the fact that
> almost all unit fractions in the sequence
> are near 0

and cannot be resolved into indivduals. ℵo will always remain as a crowd.

Regards, WM

On 1/29/2023 1:26 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 28. Januar 2023 um 23:22:26 UTC+1:
>> On 1/28/2023 5:52 AM, WM wrote:

>>> The sequence of unit fractions converges.
>>
>> Yes.
>> That means that
>> almost all unit fractions in the sequence
>> are near 0
>
> So near that they cannot be resolved into
> individuals.

Almost all of the unit fractions
in sequence ⟨ 1/1 1/2 1/3 ... ⟩ are near 0
( In each neighborhood of 0
( there is a last missed or none missed.
( There is no last in ⟨ 1/1 1/2 1/3 ... ⟩
( See also Archimedean property.

For each unit fraction 1/n in ⟨ 1/1 1/2 1/3 ... ⟩
for each split between 1/1 and 1/n
some 1/i,1/i⁺⁺ is last-before,first-after

If
there is any 1/n in ⟨ 1/1 1/2 1/3 ... ⟩ which
"cannot be resolved into an individual"
then
there is a split of ⟨ 1/1 1/2 1/3 ... ⟩
such that
1/j,1/j⁺⁺ is last-before,first-after
1/j can be resolved into an individual and
1/j⁺⁺ cannot be resolved into an individual.

If,
as seems very reasonable,
that is impossible,
for both
1/j to be resolvable into an individual
and
1/j++ to not be resolvable into an individual
then,
no,
there is no unit fraction in ⟨ 1/1 1/2 1/3 ... ⟩
-- almost all of which are near 0 --
which cannot be resolved into an individual.

>>> 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.
>>
>> All unit fractions in the sequence,
>> which means
>> all unit fractions such that,
>> for each split between 1/1 and 1/n
>> some 1/i,1/i⁺⁺ is last-before,first-after,
>> sit in an interval which ends
>> _no non-zero distance_ from 0
>
> The difference is dark.

The difference is not non-zero.
What is not non-zero? Zero.
Your Big World O' Dark is empty.

The "problems" that you (WM) "solve"
come from assuming that the infinite is finite.
Which it isn't.

>> If it were otherwise,
>> that would contradict the fact that
>> almost all unit fractions in the sequence
>> are near 0
>
> and cannot be resolved into indivduals.

For each split between 1/1 and 1/n
some 1/i,1/i⁺⁺ is last-before,first-after.

There is no first unresolvable.
There is no unresolvable at all.

> ℵo will always remain as a crowd.

"Infinite" does not mean
"reallyreallyreallyreallyreallyreally large
but finite == you can get there from here".

On Saturday, 28 January 2023 at 13:23:15 UTC-4, WM wrote:
> Gus Gassmann schrieb am Samstag, 28. Januar 2023 um 16:01:49 UTC+1:
> > On Saturday, 28 January 2023 at 06:44:30 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Freitag, 27. Januar 2023 um 20:21:39 UTC+1:
> > > > On Friday, 27 January 2023 at 06:10:04 UTC-4, WM wrote:
> > > > > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> > > > You can't do that!!!
> > > Of course I can and I did
> > You cannot,
> I did. Why did you delete the examples?

Because this SHIT does not define a distance, you fucking idiot. If you had a single neuron and the ability to read and reproduce a proper mathematical definition, you would know that.
[...] Shit deleted

> Do you have the illusion that you are right when you delete the counter evidence?

I AM right, and I did take your idiotic shit out again. It deeply offends me to have to read your crap.

Gus Gassmann schrieb am Sonntag, 29. Januar 2023 um 21:54:40 UTC+1:
> On Saturday, 28 January 2023 at 13:23:15 UTC-4, WM wrote:

> > > > > > Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.
> > > > > You can't do that!!!
> > > > Of course I can and I did
> > > You cannot,
> > I did. Why did you delete the examples?
> Because this SHIT does not define a distance

It defines a NUF.

> > Do you have the illusion that you are right when you delete the counter evidence?
> I AM right,

That feeling is insufficient in mathematics. Here you are beaten:

Inaccessible finite ordinals in the half-open intervall (0, 1]

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.

The intervall (0,1] contains all unit fractions 1/n.
Move the cursor from 1 to 0.
How many have you passed?
How many have you recognized?
There is a huge difference.
If they all were well-ordered, then you would have passed a last one because you have passed all such that no more is following. But you cannot determine which was the last one. That proves that most unit fractions are dark, i.e., not individually accessible.

Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.

For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to crover it, we have a convincing proof of dark unit fractions in the vicinity of zero.

Regards, WM

Jim Burns schrieb am Sonntag, 29. Januar 2023 um 20:30:17 UTC+1:
> On 1/29/2023 1:26 PM, WM wrote:

> > The difference is dark.
> The difference is not non-zero.

Then you need to find the last unit fraction which is responsible for the extent to zero. There is no point between 0 and (0, 1], z´the interval completely covered by unit fractions.

> The "problems" that you (WM) "solve"
> come from assuming that the infinite is finite.

The interval (0, 1] is finite. The number of points between 0 and (0, 1] is zero.
Nothing infinite as far.

> Which it isn't.

There is no point between 0 and (0, 1]. so there cannot exist a unit fractions beween them.
Everything is finite here.

> > ℵo will always remain as a crowd.
> "Infinite" does not mean
> "reallyreallyreallyreallyreallyreally large
> but finite == you can get there from here".

Zero means zero.

Inaccessible finite ordinals in the half-open intervall (0, 1]

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.

The intervall (0,1] contains all unit fractions 1/n.
Move the cursor from 1 to 0.
How many have you passed?
How many have you recognized?
There is a huge difference.
If they all were well-ordered, then you would have passed a last one because you have passed all such that no more is following. But you cannot determine which was the last one. That proves that most unit fractions are dark, i.e., not individually accessible.

Let us denote the Number of Unit Fractions existing between an object O on the positive real axis and zero as NUF_O.

For the interval [1, 2], for instance, we find NUF_[1, 2] = ℵo.
For the half-open interval (0, 1] we find NUF_(0, 1] = 0.
But for every definable unit fraction 1/n we find NUF_1/n = ℵo.

Since all unit fractions cover the interval (0, 1] but the definable unit fractions fail to cover it, we have a convincing proof of dark unit fractions in the vicinity of zero.

Regrads, WM