torsdag 17 juni 2021 kl. 11:29:24 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 17. Juni 2021 um 06:41:52 UTC+2:
> > onsdag 16 juni 2021 kl. 16:53:51 UTC+2 skrev WM:
>
> > > Resumé: Even if we don't agree about the unlimited case we can state that in our limited world there are dark numbers. Our mathematics is restricted to this world.
> > > >
> > There is nothign "dark"
>
> If Achilles passes all frames, then most of them are dark. If you deny this, then you admit that Achilles does not pass all frames. This is a silly standpoint.
>
> Regards, WM

You are making up shit with this. There are no dark numbers. What achilles do in your imaginary example is of no consequence to the fact that your idea of "dark" is completely meaningless non-sense.

William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
> On Wednesday, June 16, 2021 at 6:33:06 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 16. Juni 2021 um 19:00:11 UTC+2:
> > > The fact that you cannot write them down does not mean they have no properties,
> > If they have properties then many of these properties cannot be known.
> Maybe, but many important properties can be known ( e.g. the are still subject to induction so any property proven by induction can be known, still are the largest elements of some FISON (that can also not be written down), still have a prime decomposition, still are elements of the closed ordered field etc. )

If Achilles passes all frames, then most of them are dark. No induction till the last finitely indexed frame. If you deny this, then you admit that Achilles does not pass all frames. His life has a gap. That may be pssoible but in my opinion it is wrong.

>> But that is not enough when we remember Achilles. He has no gaps in his life line. He passes all frames.
> Correct, But he does not pass a last frame before 0.

No last natnumber can only be possible if the counting does never end. But for Achilles it has ended. Therefore there must be a last counting number.

> > If the dark frames are well-ordered, then he passes also a last finitely indexed frame.
> Nope, whether or not the "dark frames" exist, and whether or how they are ordered if they do exist does not change the fact that Achilles does not pass a last finitely indexed frame.

Then he'd remain in the counting. Otherwise this is impossible.

> > Can that be? Hardly. Therefore the dark frames may be out of order, out of linear sequence. That would be more satisfying.
> >
> > Anyhow it is nonsense to claim that Achilles passes no last finite number. This is only possible when the sequence of numbers does not end.
> Indeed, it does not end in the sense that there is always a next finite number.

It does. At frame omega there is no next finite number.

> However, there is a finite time by which all points are passed, Note that this time cannot be the time at which he passes a finite number.

Therefore the last finite numbers are dark and cannot be ordered in time or mapped to time.

> > But we know that a state appears where this sequence has ended,
> Nope, the sequence has not suddenly become a sequence with a last element (no last element has magically appeared) but Achilles has passed all infinitely many elements of the sequence. (He can do this in a finite time)

But not without passing a last one. If this last one cannot be determined, it is dark.

Regards, WM

WM explained :
> FromTheRafters schrieb am Mittwoch, 16. Juni 2021 um 23:46:12 UTC+2:
>> WM was thinking very hard :
>
>>> They are between the last natnumber connected to 1 by a FISON and omega.
>>> They are necessary to avoid a gap in the life line of Achilles.
>> News flash -- Achilles died some time ago.
>
> But Achilles passed all frames. Most of them are dark. If you deny this, then
> you admit that Achilles experienced a gap and did not pass all frames. This
> is a silly standpoint.
>
> Regards, WM

It sure is, because Achilles never 'leaves' the real number line, and
the 'frames' are infinitely many and infinitely small. The thought
experiment is not properly attacked by discrete mathematics and
calculus wins the day.

On Thursday, June 17, 2021 at 11:39:17 AM UTC+2, WM wrote:
> William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
> > On Wednesday, June 16, 2021 at 6:33:06 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 16. Juni 2021 um 19:00:11 UTC+2:
> > > > The fact that you cannot write them down does not mean they have no properties,
> > > If they have properties then many of these properties cannot be known..
> > Maybe, but many important properties can be known ( e.g. the are still subject to induction so any property proven by induction can be known, still are the largest elements of some FISON (that can also not be written down), still have a prime decomposition, still are elements of the closed ordered field etc. )
> If Achilles passes all frames, then most of them are dark. No induction till the last finitely indexed frame. If you deny this, then you admit that Achilles does not pass all frames. His life has a gap. That may be pssoible but in my opinion it is wrong.
> >> But that is not enough when we remember Achilles. He has no gaps in his life line. He passes all frames.
> > Correct, But he does not pass a last frame before 0.
>
> No last natnumber can only be possible if the counting does never end. But for Achilles it has ended. Therefore there must be a last counting number..
> > > If the dark frames are well-ordered, then he passes also a last finitely indexed frame.
> > Nope, whether or not the "dark frames" exist, and whether or how they are ordered if they do exist does not change the fact that Achilles does not pass a last finitely indexed frame.
> Then he'd remain in the counting. Otherwise this is impossible.
> > > Can that be? Hardly. Therefore the dark frames may be out of order, out of linear sequence. That would be more satisfying.
> > >
> > > Anyhow it is nonsense to claim that Achilles passes no last finite number. This is only possible when the sequence of numbers does not end.
> > Indeed, it does not end in the sense that there is always a next finite number.
> It does. At frame omega there is no next finite number.
>
> > However, there is a finite time by which all points are passed, Note that this time cannot be the time at which he passes a finite number.
>
> Therefore the last finite numbers are dark and cannot be ordered in time or mapped to time.
> > > But we know that a state appears where this sequence has ended,
> > Nope, the sequence has not suddenly become a sequence with a last element (no last element has magically appeared) but Achilles has passed all infinitely many elements of the sequence. (He can do this in a finite time)
> But not without passing a last one. If this last one cannot be determined, it is dark.
>
> Regards, WM

At once a voice arose among
The bleak twigs overhead,
In a full-hearted evensong
Of joy illimited.
An aged thrush, frail, gaunt and small,
With blast-beruffled plume,
Had chosen thus to fling his soul
Upon the growing gloom …
--Thomas Hardy

On Thursday, June 17, 2021 at 6:39:17 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:

> > ...Achilles has passed all infinitely many elements of the sequence. (He can do this in a finite time)
> But not without passing a last one.

No. Achilles passes an infinite number of points. The fact that you can select from those points a set without last element, does not mean he gets "stuck" in this set. He passes every element of the sequence despite the fact that the sequence has no last element (note he must be able to pass an infinite number of points to do this).

--
William Hughes

William schrieb am Donnerstag, 17. Juni 2021 um 15:49:24 UTC+2:
> On Thursday, June 17, 2021 at 6:39:17 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
> > > ...Achilles has passed all infinitely many elements of the sequence. (He can do this in a finite time)
> > But not without passing a last one.
> No. Achilles passes an infinite number of points. The fact that you can select from those points a set without last element, does not mean he gets "stuck" in this set. He passes every element of the sequence despite the fact that the sequence has no last element

How that? After passing every he has passed all. This implies a last one passed.

> (note he must be able to pass an infinite number of points to do this).

Yes, but every definable natnumber is a natnumber that belongs to a finite set. Without dark numbers there is no aleph_0.

Regards, WM

On 6/17/2021 5:53 AM, FromTheRafters wrote:
> WM explained :
>> FromTheRafters schrieb am Mittwoch, 16. Juni 2021 um 23:46:12 UTC+2:
>>> WM was thinking very hard :
>>
>>>> They are between the last natnumber connected to 1 by a FISON and
>>>> omega. They  are necessary to avoid a gap in the life line of Achilles.
>>> News flash -- Achilles died some time ago.
>>
>> But Achilles passed all frames. Most of them are dark. If you deny
>> this, then you admit that Achilles experienced a gap and did not pass
>> all frames. This is a silly standpoint.
>>
>> Regards, WM
>
> It sure is, because Achilles never 'leaves' the real number line, and
> the 'frames' are infinitely many and infinitely small. The thought
> experiment is not properly attacked by discrete mathematics and calculus
> wins the day.

that sums it up

William wrote on 6/17/2021 :
> On Thursday, June 17, 2021 at 6:39:17 AM UTC-3, WM wrote:
>> William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
>
>>> ...Achilles has passed all infinitely many elements of the sequence. (He
>>> can do this in a finite time)
>> But not without passing a last one.
>
> No. Achilles passes an infinite number of points.

If he ever even gets started, he instantly accomplishes this seemingly
impossible supertask.

http://wearcam.org/absement/Derivatives_of_displacement.htm

On 6/17/2021 9:30 AM, WM wrote:
> William schrieb am Donnerstag, 17. Juni 2021 um 15:49:24 UTC+2:
>> On Thursday, June 17, 2021 at 6:39:17 AM UTC-3, WM wrote:
>>> William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
>>>> ...Achilles has passed all infinitely many elements of the sequence. (He can do this in a finite time)
>>> But not without passing a last one.
>> No. Achilles passes an infinite number of points. The fact that you can select from those points a set without last element, does not mean he gets "stuck" in this set. He passes every element of the sequence despite the fact that the sequence has no last element
>
> How that? After passing every he has passed all. This implies a last one passed.

no, there is no last one.

>
>> (note he must be able to pass an infinite number of points to do this).
>
> Yes, but every definable natnumber is a natnumber that belongs to a finite set. Without dark numbers there is no aleph_0.

you use flawed logic.

all men have two legs
A monkey has two legs

therfore all men are monkeys.

fail. you always will fail with dark numbers, as they are imagination.

>
> Regards, WM
>

It happens that WM formulated :
> William schrieb am Donnerstag, 17. Juni 2021 um 15:49:24 UTC+2:
>> On Thursday, June 17, 2021 at 6:39:17 AM UTC-3, WM wrote:
>>> William schrieb am Donnerstag, 17. Juni 2021 um 02:33:58 UTC+2:
>>>> ...Achilles has passed all infinitely many elements of the sequence. (He
>>>> can do this in a finite time)
>>> But not without passing a last one.
>> No. Achilles passes an infinite number of points. The fact that you can
>> select from those points a set without last element, does not mean he gets
>> "stuck" in this set. He passes every element of the sequence despite the
>> fact that the sequence has no last element
>
> How that? After passing every he has passed all. This implies a last one
> passed.

No it doesn't. Having a last one would make it finite.

On 6/17/2021 4:29 AM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 17. Juni 2021 um 06:41:52 UTC+2:
>> onsdag 16 juni 2021 kl. 16:53:51 UTC+2 skrev WM:
>
>>> Resumé: Even if we don't agree about the unlimited case we can state that in our limited world there are dark numbers. Our mathematics is restricted to this world.
>>>>
>> There is nothign "dark"
>
> If Achilles passes all frames, then most of them are dark.

he passes all "frames", there are no dark frames.

>If you deny this,

you fail to present any Math to support your imagination.

>then you admit that Achilles does not pass all frames.

Not at all. your still counting frames, and taking time to count each
one, fail.

>This is a silly standpoint.

Agree, your standpoint is silly, and totally useless.

>
> Regards, WM
>

On 6/16/2021 4:34 PM, WM wrote:
> Greg Cunt schrieb am Mittwoch, 16. Juni 2021 um 17:12:45 UTC+2:
>> On Wednesday, June 16, 2021 at 4:53:51 PM UTC+2, WM wrote:
>>
>>> in our limited world there are dark numbers.
>> Really?! WHERE ARE they?
>
> They are between the last natnumber

there is no last natural number

On 6/16/2021 4:46 PM, FromTheRafters wrote:
> WM was thinking very hard :
>> Greg Cunt schrieb am Mittwoch, 16. Juni 2021 um 17:12:45 UTC+2:
>>> On Wednesday, June 16, 2021 at 4:53:51 PM UTC+2, WM wrote:
>>>> in our limited world there are dark numbers.
>>> Really?! WHERE ARE they?
>>
>> They are between the last natnumber connected to 1 by a FISON and
>> omega. They are necessary to avoid a gap in the life line of Achilles.
>
> News flash -- Achilles died some time ago.

the Gaps in dear departed Achilles lifeline, were caused by Ouzo.

Sergio explained :
> On 6/16/2021 4:46 PM, FromTheRafters wrote:
>> WM was thinking very hard :
>>> Greg Cunt schrieb am Mittwoch, 16. Juni 2021 um 17:12:45 UTC+2:
>>>> On Wednesday, June 16, 2021 at 4:53:51 PM UTC+2, WM wrote:
>>>>> in our limited world there are dark numbers.
>>>> Really?! WHERE ARE they?
>>>
>>> They are between the last natnumber connected to 1 by a FISON and
>>> omega. They are necessary to avoid a gap in the life line of Achilles.
>>
>> News flash -- Achilles died some time ago.
>
> the Gaps in dear departed Achilles lifeline, were caused by Ouzo.