Sergio schrieb am Dienstag, 6. Juli 2021 um 19:24:40 UTC+2:
> On 7/6/2021 12:16 PM, WM wrote:
>
> There are infinitely many. They are passed either simultaneously or one
> by one.
> None are passed simultaneously.

Then they are passed one by one.

> >In the latter case there is a last one passed.
> there are no "last" unit fractions

We consider only the unit fractions whch are fixed to points of the interval at the beginning and do not disappear during the movement of the cursor. They are *all* there.

Regards, WM

On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > In what universe is that the same as "two numbers occupy the same space"
> In our universe.

I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?

Your delusions are definitely getting worse, your statements are get more and more ridiculous, and it's been months if not years since you actually even *attempted* to write down a proof. I must assume that you are finally realizing that you simply are not up to the task, definitely not now, and probably at no point in your miserable existence. You are a lying, deceitful crank, and you are losing your marbels in front of everyone to see.

On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 18:32:01 UTC+2:
> > On Tuesday, July 6, 2021 at 11:59:53 AM UTC-4, WM wrote:
> > > William schrieb am Montag, 5. Juli 2021 um 21:51:35 UTC+2:
> > > > : When the cursor reaches 0 no unit fraction is between itself and 0. And fact is further that the cursor never passes aleph_0 unit fractions at the same point.
> > > >
> > > > Correct. So what?
> > > It excludes the claim that for all cursor positions > 0 there are aleph_0 unit fractions between itself and zero
> > No it doesn't.
> >
> > >Then the cursor would have to pass aleph_0 elements of U_F between (0, 1] and 0.
> >
> > Nope, it is sufficient that the cursor passes aleph_0 elements of U_F in any neighborhood of 0. Of course you get a contradiction at the last neighborhood, Guess What?
> There is a last position, it is zero and it exists.

0 is not a neighborhood of 0.

William schrieb am Dienstag, 6. Juli 2021 um 19:57:14 UTC+2:
> On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:

> > There is a last position, it is zero and it exists.
> 0 is not a neighborhood of 0.

Every neighbourhood that can be defined contains aleph_0 unit fractions. That is true. But When the cursor arrives at zero it has passed all such that none remains. Can this have happended simultaneously or only one by one? That is the question.

Regards, WM

Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 19:55:05 UTC+2:
> On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > In what universe is that the same as "two numbers occupy the same space"
> > In our universe.
> I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?

If between cursor and zero there are aleph_0 unit fractions at every point in (0, 1] but none at zero, then all have been sitting between (0, 1] and 0.

When the cursor arrives at zero it has passed all such that none remains. Can this have happended simultaneously or only one by one? That is the question.

Regards, WM

On Tuesday, July 6, 2021 at 1:45:55 PM UTC-4, WM wrote:
> Sergio schrieb am Dienstag, 6. Juli 2021 um 19:24:40 UTC+2:
> > On 7/6/2021 12:16 PM, WM wrote:
> >
> > There are infinitely many. They are passed either simultaneously or one
> > by one.
> > None are passed simultaneously.
>
> Then they are passed one by one.

Correct

> In [this] case there is a last one passed.

Nope. This does not follow and is not true.

--
William Hughes

On Tuesday, July 6, 2021 at 2:15:23 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 19:57:14 UTC+2:
> > On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:
>
> > > There is a last position, it is zero and it exists.
> > 0 is not a neighborhood of 0.
> Every neighbourhood that can be defined

is a set of neighborhoods with last element and irrelevant.

--
William Hughes

On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 19:55:05 UTC+2:
> > On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > > In what universe is that the same as "two numbers occupy the same space"
> > > In our universe.
> > I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?
> If between cursor and zero there are aleph_0 unit fractions at every point in (0, 1] but none at zero, then

every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1].. (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")

--
William Hughe

On Tuesday, 6 July 2021 at 15:17:52 UTC-3, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 19:55:05 UTC+2:
> > On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > > In what universe is that the same as "two numbers occupy the same space"
> > > In our universe.
> > I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?
> If between cursor and zero there are aleph_0 unit fractions at every point in (0, 1] but none at zero, then all have been sitting between (0, 1] and 0.

No, you dumb shit, you swapped quantifiers. For every point x greater than 0 there are infinitely many points in the interval (0,x).

On Tuesday, 6 July 2021 at 13:45:55 UTC-4, WM wrote:
> Sergio schrieb am Dienstag, 6. Juli 2021 um 19:24:40 UTC+2:
> > On 7/6/2021 12:16 PM, WM wrote:
> >
> > There are infinitely many. They are passed either simultaneously or one
> > by one.
> > None are passed simultaneously.
>
> Then they are passed one by one.
> > >In the latter case there is a last one passed.
> > there are no "last" unit fractions
> We consider only the unit fractions whch are fixed to points of the interval at the beginning and do not disappear during the movement of the cursor. They are *all* there.

Except for the shy ones that can't be known? LMAO.

>
> Regards, WM

On 7/6/2021 1:17 PM, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 19:55:05 UTC+2:
>> On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
>>>> On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
>>>>> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
>>>>>> On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
>>>>> Note that never two unit fractions reside at the same position, let alone aleph_0.
>>>>>> Who, other than you, have ever said or thought that different numbers could occupy the same space?
>>>>> Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
>>>> In what universe is that the same as "two numbers occupy the same space"
>>> In our universe.
>> I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?
>
> If between cursor and zero there are aleph_0 unit fractions at every point in (0, 1] but none at zero, then all have been sitting between (0, 1] and 0.

that is because you used the "(" character, do not do that again!

>
> When the cursor arrives at zero it has passed all such that none remains. Can this have happended simultaneously or only one by one? That is the question.

there is no question.

take a point, say 10, what is the point "next to" 10 ?

>
> Regards, WM
>

On 7/6/2021 11:12 AM, WM wrote:
> Jim Burns schrieb am Montag, 5. Juli 2021 um 22:00:13 UTC+2:
>> On 7/4/2021 9:08 AM, WM wrote:
>
>>> None remains, not even the infinitely many following upon
>>> every accessible unit fraction.
>>
>> Any point x before 0 and after each fraction 1/k corresponds to
>> a point 1/x after each (finite) natural k. Archimedes would say
>> that no such 1/x exist in the line, but that each point must be
>> able to be encompassed by finitely-many unit-interval extensions
>> of the line.
>>
>> No 1/x, no x, no dark points.
>
> Assume that for every point 1/x in the interval (0, 1] there are aleph_0 unit fractions between cursor and 0. In 0 there all are lost. That means all get lost between (0, 1] and 0. That is impossible. Each unit fraction occupies its own place. They cannot get lost between (0, 1] and 0. The get lost one by one - until the last one is lost. But we cannot prove that it is the last one.
>>
>>> They are not accessible. They are dark.
>
>> But you don't say anything about what your dark points are like.
>
> They are unit fractions with natural denominators. Each natural has two direct neighbours, probably many predecessors and successors, but no connection by a FISON to zero.
>
>> The only use I see for dark points is that they give you (WM)
>> something to say to impertinent students who expect you to
>> know something about the material you're lecturing on.
>
> I have no impertinent students.

they are potentially misinformed, or completed misinformed students.

>>
>> ----
>> A better description of these points is
>> (i)
>> definable points passed in every movement
>> ("definable" might only be "finite-length decimals"),
>
> Also many irrationals are defined, like pi, e, L. They all are surrounded by dark points. But in oder to keep things easy I have only used dark points before zero.
>
> Regards, WM
>

WM laid this down on his screen :
> FromTheRafters schrieb am Montag, 5. Juli 2021 um 23:29:54 UTC+2:
>> WM formulated on Monday :
>
>>> We should forget the time.
>> Then what does 'arrive' and 'passing' even mean? Time is implicit in
>> your adopting a step by step method.
>
> Only mode of speaking. "Passing" means being at the same point.

Merely a coincidence, I'm sure.

Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 20:47:46 UTC+2:
> On Tuesday, 6 July 2021 at 15:17:52 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 19:55:05 UTC+2:
> > > On Tuesday, 6 July 2021 at 13:28:06 UTC-3, WM wrote:
> > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > > > In what universe is that the same as "two numbers occupy the same space"
> > > > In our universe.
> > > I ask again: How the *FUCK* is "there are aleph_0 unit fractions between zero and the cursor [at any point in] (0,1]" the same as "two numbers occupy the same space"?
> > If between cursor and zero there are aleph_0 unit fractions at every point in (0, 1] but none at zero, then all have been sitting between (0, 1] and 0.
> you swapped quantifiers.

I do not use quantifiers. I use the unit fractions which sit in the interval (0, 1] and remain there at rest. They are passed by the cursor. Never two or more are passed. Hence there is a last one passed.

> For every point x greater than 0 there are infinitely many points in the interval (0,x).

When the cursor passes them, either they are passed simultaneopusly or one by one. No quantifiers used.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 20:20:16 UTC+2:
> On Tuesday, July 6, 2021 at 1:45:55 PM UTC-4, WM wrote:
> > Sergio schrieb am Dienstag, 6. Juli 2021 um 19:24:40 UTC+2:
> > > On 7/6/2021 12:16 PM, WM wrote:
> > >
> > > There are infinitely many. They are passed either simultaneously or one
> > > by one.
> > > None are passed simultaneously.
> >
> > Then they are passed one by one.
> Correct
>
> > In [this] case there is a last one passed.
>
> Nope. This does not follow and is not true.

It does follow and is true: Either more than one together or one by one. There is no other alternative. Note that we only consider the existing unit fractions which are there from the beginning. No not existing not last ones.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 20:25:42 UTC+2:
> On Tuesday, July 6, 2021 at 2:15:23 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 6. Juli 2021 um 19:57:14 UTC+2:
> > > On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:
> >
> > > > There is a last position, it is zero and it exists.
> > > 0 is not a neighborhood of 0.
> > Every neighbourhood that can be defined
> is a set of neighborhoods with last element and irrelevant.

When the cursor arrives at zero it has passed all unit fractions such that none remains. Can this have happended simultaneously or only one by one? That is the question.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:

> every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]

that can be defined. But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones simultaneously.

> (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")

Then the cursor has passed aleph_0 unit fractions simultaneously between (0, 1] and 0 siunce there is no dark point next to 0 but only such x as you describe. But that is a contradiction because the unit fractions sit at different points.

Regards, WM

On 7/6/2021 3:09 PM, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
>> On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
>
>
>> every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
>
> that can be defined. But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones simultaneously.

wrong, the cursor defines each number it passes, all numbers in the
interval are defined by the cursor as it moves across.

>
>> (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")
>
> Then the cursor has passed aleph_0 unit fractions simultaneously between (0, 1] and 0 siunce there is no dark point next to 0 but only such x as you describe. But that is a contradiction because the unit fractions sit at different points.

there is no contradiction. You use failed logic.

you cannot even "define" a single number.

You require listing ALL its digits

>
> Regards, WM
>

On Tuesday, July 6, 2021 at 4:04:08 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 20:25:42 UTC+2:
> > On Tuesday, July 6, 2021 at 2:15:23 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 6. Juli 2021 um 19:57:14 UTC+2:
> > > > On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:
> > >
> > > > > There is a last position, it is zero and it exists.
> > > > 0 is not a neighborhood of 0.
> > > Every neighbourhood that can be defined
> > is a set of neighborhoods with last element and irrelevant.
> When the cursor arrives at zero it has passed all unit fractions such that none remains. Can this have happended simultaneously or only one by one?

The elements of U_F are passed one by one. (trivial and stated many times) So What? The fact that the elements are passed one by one does not mean there is a last element passed.

On Tuesday, July 6, 2021 at 4:09:59 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> > On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
>
>
> > every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
> that can be defined.

Nope, any real number in (0,1] whether or not you can write it down.

> But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones
> simultaneously.

Nope the elements are passed one by one, but there is no last element that is passed, The cursor does not have to pass a last element of U_F to get to 0.

> > (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")
> Then the cursor has passed aleph_0 unit fractions simultaneously between (0, 1] and 0
..
Nope. We can only conclude that in every neighborhood of 0 the cursor has passed (one by one) an infinite number of elements of U_F.

--
William Hughes

William schrieb am Dienstag, 6. Juli 2021 um 22:52:58 UTC+2:
> On Tuesday, July 6, 2021 at 4:09:59 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> > > On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
> >
> >
> > > every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
> > that can be defined.
> Nope, any real number in (0,1] whether or not you can write it down.
> > But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones
> > simultaneously.
> Nope the elements are passed one by one, but there is no last element that is passed, The cursor does not have to pass a last element of U_F to get to 0.
> > > (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")
> > Then the cursor has passed aleph_0 unit fractions simultaneously between (0, 1] and 0
> .
> Nope. We can only conclude that in every neighborhood of 0 the cursor has passed (one by one) an infinite number of elements of U_F.

One of them was the last unless aleph_0 simultaneously have been passed last. Note that the scale from 1 to 0 implies an order.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 22:52:58 UTC+2:
> On Tuesday, July 6, 2021 at 4:09:59 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> > > On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
> >
> >
> > > every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
> > that can be defined.
> Nope, any real number in (0,1] whether or not you can write it down.
> > But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones
> > simultaneously.
> Nope the elements are passed one by one, but there is no last element that is passed, The cursor does not have to pass a last element of U_F to get to 0.

It is a matter of fact that among a sequence which is finished there is a last term.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 22:40:52 UTC+2:
> On Tuesday, July 6, 2021 at 4:04:08 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 6. Juli 2021 um 20:25:42 UTC+2:
> > > On Tuesday, July 6, 2021 at 2:15:23 PM UTC-4, WM wrote:
> > > > William schrieb am Dienstag, 6. Juli 2021 um 19:57:14 UTC+2:
> > > > > On Tuesday, July 6, 2021 at 1:20:28 PM UTC-4, WM wrote:
> > > >
> > > > > > There is a last position, it is zero and it exists.
> > > > > 0 is not a neighborhood of 0.
> > > > Every neighbourhood that can be defined
> > > is a set of neighborhoods with last element and irrelevant.
> > When the cursor arrives at zero it has passed all unit fractions such that none remains. Can this have happended simultaneously or only one by one?
> The elements of U_F are passed one by one. (trivial and stated many times) So What? The fact that the elements are passed one by one does not mean there is a last element passed.

This could be stated for an unfinished task. But the sequence of unit fractions has been finished. Hence there must have been a last event.

Regards, WM

On Tuesday, July 6, 2021 at 5:04:29 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 22:52:58 UTC+2:
> > On Tuesday, July 6, 2021 at 4:09:59 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> > > > On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
> > >
> > >
> > > > every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
> > > that can be defined.
> > Nope, any real number in (0,1] whether or not you can write it down.
> > > But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones
> > > simultaneously.
> > Nope the elements are passed one by one, but there is no last element that is passed, The cursor does not have to pass a last element of U_F to get to 0.
> > > > (Recall, that despite the misleading name, "dark real numbers" are not real numbers, so (0,1] does not contain a "dark real number")
> > > Then the cursor has passed aleph_0 unit fractions simultaneously between (0, 1] and 0
> > .
> > Nope. We can only conclude that in every neighborhood of 0 the cursor has passed (one by one) an infinite number of elements of U_F.
> One of them was the last

Nope. There is no last point at which an element of U_F is passed,

--
William Hughes

On Tuesday, July 6, 2021 at 5:06:18 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 22:52:58 UTC+2:
> > On Tuesday, July 6, 2021 at 4:09:59 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 6. Juli 2021 um 20:35:19 UTC+2:
> > > > On Tuesday, July 6, 2021 at 2:17:52 PM UTC-4, WM wrote:
> > >
> > >
> > > > every neighborhood, d, of 0 must contain an infinite subset S(d) of U_F. It is trivial to show that this is sufficient for there being aleph_0 elements of U_F between 0 and every real number in the set of real numbers (0,1]
> > > that can be defined.
> > Nope, any real number in (0,1] whether or not you can write it down.
> > > But obviously not all can be defined, because when the cursor has passed all unit fractions, then it has passed either a last one or several last ones
> > > simultaneously.
> > Nope the elements are passed one by one, but there is no last element that is passed, The cursor does not have to pass a last element of U_F to get to 0.
> It is a matter of fact that among a sequence which is finished there is a last term.

Nope, You can have a sequence which contains every element of the sequence (i.e. is finished) but does not have a last term.

--
William Hughes