On Sunday, February 19, 2023 at 8:44:38 PM UTC+1, WM wrote:

> If there are discrete points (i.e., points which are separated from each other) on a line which has an end, then there is a last point before this end.

Nope. Consider the points 1/n with n e IN, or rahter the sequence of points (p_n)_(n e IN) defined with p_n = 1/n (for all n e IN). This sequence is monotonic: An e IN: p_(n+1) < p_n.

Let's assume that 0 is "the end" of the line {x e IR : 0 <= x}. Since (p_n) does not have "a last element" (and for all i,j; i =/= j -> p_i =/= p_k) there is no "last point before this end".

Fritz Feldhase schrieb am Sonntag, 19. Februar 2023 um 20:50:21 UTC+1:
> On Sunday, February 19, 2023 at 8:33:07 PM UTC+1, WM wrote:
> > Fritz Feldhase schrieb am Sonntag, 19. Februar 2023 um 18:25:14 UTC+1:
>
> > If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> Nope. Consider the points 1/n with n e IN, or rahter the sequence of points (p_n)_(n e IN) defined with p_n = 1/n (for all n e IN). This sequence is monotonic: An e IN: p_(n+1) < p_n.
>
> Let's assume that 0 is "the end" of the line {x e IR : 0 <= x}. Since (p_n) does not have "a last element" (and for all i,j; i =/= j -> p_i =/= p_k) there is no "last point before the end".

This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.

If none of them were dark, then all could be searched and the one next to th.e end could be used.

Compare the imposibility to descend from omega to zero by passing the natural numbers. Although all are finite, it is not possible to pass all. You have to jump to a chosen one. But you can always find a larger one to choose. Why can't you pass all finite numbers? Because they are dark.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 19. Februar 2023 um 20:53:03 UTC+1:
> On Sunday, February 19, 2023 at 8:44:38 PM UTC+1, WM wrote:
>
> > If there are discrete points (i.e., points which are separated from each other) on a line which has an end, then there is a last point before this end.
> Nope. Consider the points 1/n with n e IN, or rahter the sequence of points (p_n)_(n e IN) defined with p_n = 1/n (for all n e IN). This sequence is monotonic: An e IN: p_(n+1) < p_n.
> Let's assume that 0 is "the end" of the line {x e IR : 0 <= x}. Since (p_n) does not have "a last element" (and for all i,j; i =/= j -> p_i =/= p_k) there is no "last point before this end".

Of course there is a last point. The above principle, an unshakeable truth, proves it. But it is dark.

Look, there are ℵo unit fractions between zero and the next chosable unit fraction. Why can't you choose one of these ℵo unit fractions?

Regards, WM

WM used his keyboard to write :
> Fritz Feldhase schrieb am Sonntag, 19. Februar 2023 um 18:25:14 UTC+1:
>> On Sunday, February 19, 2023 at 5:55:20 PM UTC+1, Fritz Feldhase wrote:
>>
>>> In an appropriate Gedankenexperiment the cursor does not pass "a last unit
>>> fraction" when it's "moving" from 1 to 0. (Hint: There is no smallest unit
>>> fraction in (0, 1]. _That is fact._)
>>
>> Some preconditions:
>
> Thank you for this analysis. It is unnecessary. We need no cursor. If there
> are points, which are separated from each other by non-zero distances, on a
> line which has an end, then there is a last point before this end.

What makes you think so? Between any two distinct real number points,
there are uncountably many other real number points and countably many
of them are rational.

Fritz Feldhase schrieb am Sonntag, 19. Februar 2023 um 18:25:14 UTC+1:
> On Sunday, February 19, 2023 at 5:55:20 PM UTC+1, Fritz Feldhase wrote:
>
> > In an appropriate Gedankenexperiment the cursor does not pass "a last unit fraction" when it's "moving" from 1 to 0. (Hint: There is no smallest unit fraction in (0, 1]. _That is fact._)
>
> Some preconditions:
>
> 1. The cursor has the "size" of a point.
> 2. It's moving at a constant speed. Say it moves in 1 second from point 1 to point 0. Hence v = 1/1 = 1.
> 3. "Space" and "time" can be modeled by IR. (Hence we can say that the cursor moves "on the real number line").
>
> This means that for each and every t e [0, 1] the cursor is at a certain location (point), namely x(t) = (1 - t) e IR. You see, it starts at t = 0 at point x = 1 and reaches point x = 0 at t = 1. #

Of course. All this is what I have assumed. I did not mention it, because it is taken for granted in physics.

> Now we can _prove_ that _the cursor does not pass "a last unit fraction" when he's "moving" from 1 to 0_:
>
> Assume that the cursor passes a last unit fraction. Say u > 0.

It does. If all were visible, the first one could e seen. But they cannot be discerned. The reason is their darkness. The proof of their existence is this: ℵo unit fractions are distributed at the LHS of every visible unit fraction. They are there, have distances between them, but cannot be used as individuals. Otherwise we would do it.

> Then there's a natural number n such that u = 1/n [where n =/= 1]. That means that it passes that unit fraction at time t = 1 - 1/n, where 0 < t < 1. But at time t' = 1/(1/(1 - t) + 1), where 0 < t' < t, it passes the the unit fraction 1/(n+1) which lies between 0 and u. Contradiction!

All n that can be inserted here are visible. But most cannot be inserted because they are dark.

Note that every descending way from omega to zero has a gap because after omega there is a leap to a visible number n which has ℵo successors which could not be used. Otherwise we had done it.

> Hence our assumption was false and the cursor does not pass "a last unit fraction".

Same with the way from omega to n. When you have reached n, you can return a good deal to n^n^n or comparable numbers, which however are also only tiny numbers compared to omega and the dark natural numbers immediately before omega. This holds for every visible natural number, hecause they form a potentially infinite collection.

The existence of dark numbers is guaranteed by the axiom of foundation already: Every sequence of natural numbers descending from ω to 0 is finite. But above all it is dictated by the practical impossibility to identify actually infinitely many predecessors of ω and to choose them as destinations for the leap from ω

Regards, WM

FromTheRafters schrieb am Montag, 20. Februar 2023 um 00:13:50 UTC+1:
> WM used his keyboard to write :

> > Thank you for this analysis. It is unnecessary. We need no cursor. If there
> > are points, which are separated from each other by non-zero distances, on a
> > line which has an end, then there is a last point before this end.

> What makes you think so?

The completeness of all points which are existing all the time .

> Between any two distinct real number points,
> there are uncountably many other real number points

They are there from the beginning - according to set theory. Hence there is a next one to the end.

> and countably many of them are rational.

In fact here we are concerned with unit fractions only. They are not only countable but they are countable in their natural order.

Likewise every sequence of natural numbers descending from ω to 0 is finite. This follows from the axiom of foundation.

Regards, WM

WM used his keyboard to write :
> FromTheRafters schrieb am Montag, 20. Februar 2023 um 00:13:50 UTC+1:
>> WM used his keyboard to write :
>
>>> Thank you for this analysis. It is unnecessary. We need no cursor. If there
>>> are points, which are separated from each other by non-zero distances, on a
>>> line which has an end, then there is a last point before this end.
>
>> What makes you think so?
>
> The completeness of all points which are existing all the time .

Time again? How does time fit on the number line? You misunderstand
completeness again.

On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:

> This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.

Far from an unshakeable truth, this statement is simply false.

William schrieb am Montag, 20. Februar 2023 um 16:13:51 UTC+1:
> On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:
>
> > This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> Far from an unshakeable truth, this statement is simply false.

If you move from zero to 1 you will not encounter a first unit fraction? When you see the first one, you will have encountered infinity many invisible unit fractions already.

Every unit visible fraction of U_P can be removed by subtracting it. Every unit fraction of U_P has ℵo unit fradtions at its LHS, ℵo of which cannot be removed by subtracting them individually.

Regards, WM

FromTheRafters schrieb am Montag, 20. Februar 2023 um 10:25:09 UTC+1:
> WM used his keyboard to write :
> > FromTheRafters schrieb am Montag, 20. Februar 2023 um 00:13:50 UTC+1:
> >> WM used his keyboard to write :
> >
> >>> Thank you for this analysis. It is unnecessary. We need no cursor. If there
> >>> are points, which are separated from each other by non-zero distances, on a
> >>> line which has an end, then there is a last point before this end.
> >
> >> What makes you think so?
> >
> > The completeness of all points which are existing all the time .
> Time again?

No. All the time invariable means no time dependence.

Regards, WM

WM pretended :
> FromTheRafters schrieb am Montag, 20. Februar 2023 um 10:25:09 UTC+1:
>> WM used his keyboard to write :
>>> FromTheRafters schrieb am Montag, 20. Februar 2023 um 00:13:50 UTC+1:
>>>> WM used his keyboard to write :
>>>>> Thank you for this analysis. It is unnecessary. We need no cursor. If
>>>>> there are points, which are separated from each other by non-zero
>>>>> distances, on a line which has an end, then there is a last point before
>>>>> this end.
>>>> What makes you think so?
>>>
>>> The completeness of all points which are existing all the time .
>> Time again?
>
> No. All the time invariable means no time dependence.

Why not say "The completeness of all points which are existing" and
leave the waffle for breakfast?

On Monday, February 20, 2023 at 10:33:15 PM UTC+1, WM wrote:

> If you move from zero to 1 you will not encounter a first unit fraction?

No, you won't.

Hint: At t = 0 you haven't yet "encountered" a unit fraction. At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions. So there's no first you will/can "encounter".

Need a proof for that? (But you may check and adopt my analysis of your recent Gedankenmodel yourself.)

> When <bla>

On Monday, February 20, 2023 at 5:33:15 PM UTC-4, WM wrote:
> William schrieb am Montag, 20. Februar 2023 um 16:13:51 UTC+1:
> > On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:
> >
> > > This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> > Far from an unshakeable truth, this statement is simply false.
> If you move from zero to 1 you will not encounter a first unit fraction

Correct. Pick any any element (each element of U_P is "pickable") and move to it. You will pass an infinite number of unit elements of U_P each one of which is "pickable". There is no first element of U_P that is "pickeable".

Fritz Feldhase schrieb am Montag, 20. Februar 2023 um 23:27:49 UTC+1:
> On Monday, February 20, 2023 at 10:33:15 PM UTC+1, WM wrote:
>
> > If you move from zero to 1 you will not encounter a first unit fraction?
> No, you won't.
>
> Hint: At t = 0 you haven't yet "encountered" a unit fraction. At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions..

Here we have to distinguish two different meanings of to encounter or to pass. When you see (consciously encounter or pass) the first unit fraction, then you have passed already ℵo-infinitely many unit fractions, many of which you can return to.

> So there's no first you will/can "encounter".

Since all unit fractions have distances, there is a first to pass and a first to be seen. Compare the descending sequence from ω to 0 which is finite but passes ℵo natural numbers, that means you pass infinitely many but can see only finitely many (axiom of foundation).

Regards, WM
>

William schrieb am Montag, 20. Februar 2023 um 23:28:21 UTC+1:
> On Monday, February 20, 2023 at 5:33:15 PM UTC-4, WM wrote:
> > William schrieb am Montag, 20. Februar 2023 um 16:13:51 UTC+1:
> > > On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:
> > >
> > > > This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> > > Far from an unshakeable truth, this statement is simply false.
> > If you move from zero to 1 you will not encounter a first unit fraction
> Correct. Pick any any element (each element of U_P is "pickable")

But not its ℵo predecessors which hiwever have to be passed, one by one, before you can pick a unit fraction.

> You will pass an infinite number of unit elements of U_P each one of which is "pickable".

Yes. It is like the descending sequence from ω to 0 which is finite but passes ℵo natural numbers, that means you pass infinitely many but can see only finitely many (axiom of foundation).

You cannot pick one with less predecessors. This shows that every picked one has dark predecessors.

> There is no first element of U_P that is "pickeable".

Right. The collection is potentially infinite.

There are all unit fractions existing in actual infinity. None is constructed. Only such can be chosen, which have infinitely many dark unit fractions at their LHS. Hence infinitely many cannot be chosen.

Regards, WM

On Tuesday, February 21, 2023 at 4:50:52 AM UTC-4, WM wrote:
> William schrieb am Montag, 20. Februar 2023 um 23:28:21 UTC+1:
> > On Monday, February 20, 2023 at 5:33:15 PM UTC-4, WM wrote:
> > > William schrieb am Montag, 20. Februar 2023 um 16:13:51 UTC+1:
> > > > On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:
> > > >
> > > > > This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> > > > Far from an unshakeable truth, this statement is simply false.
> > > If you move from zero to 1 you will not encounter a first unit fraction
> > Correct. Pick any any element (each element of U_P is "pickable")
> But not its ℵo predecessors

Nope. Each predecessor is "pickable".

<snip>

> The collection is potentially infinite.

Nope. No matter how many times you say this it is false. U_P is a set, it does not change.

>
> There are all [elements of U_P] existing in actual infinity. None is constructed. Only such can be chosen,
> which have infinitely many dark unit fractions at their LHS.

False. Each element can be chosen. U_P does not have any "dark" elements

On Tuesday, February 21, 2023 at 9:50:46 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Montag, 20. Februar 2023 um 23:27:49 UTC+1:
> > On Monday, February 20, 2023 at 10:33:15 PM UTC+1, WM wrote:
> > >
> > > If you move from zero to 1 you will not encounter a first unit fraction?
> > >
> > No, you won't.
> >
> > Hint: At t = 0 you haven't yet "encountered" a unit fraction. At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions.
> >
> Here we have <bla>

No, we haven't.

Hint:

> > So there's no first you will/can "encounter".
> >
> Since all unit fractions have distances, there is a first to pass [...]

Nope. Non sequitur.

Hint: These distances tend to 0 (for n -> oo) and there is no first "to pass" (see comment from above).

> you pass infinitely many but can see only finitely many (axiom of foundation).

Actually, I can not even see ONE unit fraction, idiot. Moreover: Has NOTHING to do with the _axiom of foundation_, silly.

See: https://en.wikipedia.org/wiki/Axiom_of_regularity

William schrieb am Dienstag, 21. Februar 2023 um 16:33:46 UTC+1:
> On Tuesday, February 21, 2023 at 4:50:52 AM UTC-4, WM wrote:
> > William schrieb am Montag, 20. Februar 2023 um 23:28:21 UTC+1:
> > > On Monday, February 20, 2023 at 5:33:15 PM UTC-4, WM wrote:
> > > > William schrieb am Montag, 20. Februar 2023 um 16:13:51 UTC+1:
> > > > > On Sunday, February 19, 2023 at 3:56:18 PM UTC-4, WM wrote:
> > > > >
> > > > > > This is an unshakeable truth: If there are points, which are separated from each other by non-zero distances, on a line which has an end, then there is a last point before this end.
> > > > > Far from an unshakeable truth, this statement is simply false.
> > > > If you move from zero to 1 you will not encounter a first unit fraction
> > > Correct. Pick any any element (each element of U_P is "pickable")
> > But not its ℵo predecessors
> Nope. Each predecessor is "pickable".
>
> > The collection is potentially infinite.
> Nope. No matter how many times you say this it is false. U_P is a set, it does not change.

Then find the element which is closest to zero. Infinitely many invarible points on the real axis with internal distances of infinitely many points each do not fit between zero and the closest x > 0.
>
> >
> > There are all [elements of U_P] existing in actual infinity. None is constructed. Only such can be chosen,
> > which have infinitely many dark unit fractions at their LHS.
> False. Each element can be chosen. U_P does not have any "dark" elements

All positive points x which are subject to the theorem (that there is a distance from zero which contains ℵo unit fractions and their internal distances), have a distance from zero which contains ℵo unit fractions and their internal distances. Closer points cannot be chosen, but are existing.

Regrads, WM

Fritz Feldhase schrieb am Dienstag, 21. Februar 2023 um 17:10:21 UTC+1:
> On Tuesday, February 21, 2023 at 9:50:46 AM UTC+1, WM wrote:
> > Fritz Feldhase schrieb am Montag, 20. Februar 2023 um 23:27:49 UTC+1:
> > > On Monday, February 20, 2023 at 10:33:15 PM UTC+1, WM wrote:
> > > >
> > > > If you move from zero to 1 you will not encounter a first unit fraction?
> > > >
> > > No, you won't.
> > >
> > > Hint: At t = 0 you haven't yet "encountered" a unit fraction. At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions.
> > >
> > All points on the real line which have internal distances from each other can be distinguished if they are not dark.

> Hint: These distances tend to 0 (for n -> oo) and there is no first "to pass" (see comment from above).

None of the distances contains less than infinitely many points.

> > you pass infinitely many but can see only finitely many (axiom of foundation).
> Actually, I can not even see ONE unit fraction

Then try to get a better training. For you everything must be dark.

> Moreover: Has NOTHING to do with the _axiom of foundation_, silly.

Of course. This axiom simulates dark elements.

Why can't you distingusih all natural numbers when descendig? Because they are dark. You can't distinguish them either when ascending. It only appears so by "and so on" .

Regards, WM

On Tuesday, February 21, 2023 at 4:08:33 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 21. Februar 2023 um 16:33:46 UTC+1:
> > On Tuesday, February 21, 2023 at 4:50:52 AM UTC-4, WM wrote:

> > > The collection is potentially infinite.
> > Nope. No matter how many times you say this it is false. U_P is a set, it does not change.
> Then find the element which is closest to zero.

There is of course no such element. For each element of U_P, x, there is a set, S(x), of elements of U_P between x and 0. The sets S(x) do not change.

Fritz Feldhase schrieb am Dienstag, 21. Februar 2023 um 17:10:21 UTC+1:

> > > At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions.

Obviously they cannot be distinguished. You cannot choose one of them such that there are less than ℵo between it and zero. But these ℵo are existing because you have encountered them on the real line .

Regards, WM

William schrieb am Dienstag, 21. Februar 2023 um 21:23:46 UTC+1:
> On Tuesday, February 21, 2023 at 4:08:33 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 21. Februar 2023 um 16:33:46 UTC+1:
> > > On Tuesday, February 21, 2023 at 4:50:52 AM UTC-4, WM wrote:
>
> > > > The collection is potentially infinite.
> > > Nope. No matter how many times you say this it is false. U_P is a set, it does not change.
> > Then find the element which is closest to zero.
> There is of course no such element.

But there are ℵo unit fractions which you will pass when starting at zero and going to any choosable unit fraction.

> For each element of U_P, x, there is a set, S(x), of elements of U_P between x and 0. The sets S(x) do not change.

But you cannot choose one of the first ℵo elements which are closest to zero, closer than any element that you can choose. They are points on the real axis. They cannot be chosen.

Regards, WM.

On Tuesday, February 21, 2023 at 4:28:38 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 21. Februar 2023 um 21:23:46 UTC+1:
> > On Tuesday, February 21, 2023 at 4:08:33 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 21. Februar 2023 um 16:33:46 UTC+1:
> > > > On Tuesday, February 21, 2023 at 4:50:52 AM UTC-4, WM wrote:
> >
> > > > > The collection is potentially infinite.
> > > > Nope. No matter how many times you say this it is false. U_P is a set, it does not change.
> > > Then find the element which is closest to zero.
> > There is of course no such element.
> But there are ℵo unit fractions which you will pass when starting at zero and going to any choosable unit fraction.

Correct and each of them is "choosable" (your latest word for "has the property that it can be written down").

On Tuesday, February 21, 2023 at 9:24:02 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 21. Februar 2023 um 17:10:21 UTC+1:
> > > >
> > > > At _any_ t e IR, t > 0, you already have "encoutered" infinitely many unit fractions.
> > > >
> Obviously they cannot be distinguished.

Was immer das auch heißen soll, Mückenheim.

On Tuesday, February 21, 2023 at 9:15:22 PM UTC+1, WM wrote:

> [The] axiom [of foundation] simulates dark elements.

Yeah, whatever, Mückenheim.

EOD