On Sunday, February 12, 2023 at 12:35:19 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 12. Februar 2023 um 17:28:50 UTC+1:
> > On Sunday, February 12, 2023 at 12:16:37 PM UTC-4, WM wrote:
>
> > > No. |N_P is a potentially infinite collection, always finite but not fixed
> > Nope, |N_P is a set. It does not change.
> What about the unit fractions that cannot be chosen?

"dark unit fractions" are larger than each element of the set |N_P and are not elements of |R

WM pretended :
> FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 14:27:39 UTC+1:
>
> Between every x > 0 that can be chosen

Explain "can be chosen" in this context. Is it the same as "can be
distinguished" "can be discerned" "can be written down" "can be counted
out loud by a horse hoof stomping algorithm" etcetera?

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

WM explained on 2/12/2023 :
> FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 14:27:39 UTC+1:
>
> Between every x > 0 that can be chosen and 0 there are ℵ₀ unit fractions as
> geometric points (an in addition they have intervals between them).

Between *each* x > 0 and zero.

> THIS IMPLIES Between all x > 0

No, *each* (every individual such element) is not the same as *all*
(the set of such elements).

On Sunday, February 12, 2023 at 12:35:19 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 12. Februar 2023 um 17:28:50 UTC+1:
> > On Sunday, February 12, 2023 at 12:16:37 PM UTC-4, WM wrote:
>
> > > No. |N_P is a potentially infinite collection, always finite but not fixed
> > Nope, |N_P is a set. It does not change.
> What about the unit fractions that cannot be chosen?

They are reciprocals of "dark natural numbers" which are larger than each element of |N_P. Neither the "dark natural numbers" nor their reciprocals are elements of |R.

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 17:54:54 UTC+1:
> WM pretended :

> > Between every x > 0 that can be chosen
> Explain "can be chosen" in this context.

Describe it so that a mathematician can know what you have described.

Regards, WM

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 18:12:03 UTC+1:
> WM explained on 2/12/2023 :
> > FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 14:27:39 UTC+1:
> >
> > Between every x > 0 that can be chosen and 0 there are ℵ₀ unit fractions as
> > geometric points (an in addition they have intervals between them).
> Between *each* x > 0 and zero.

Not only that. Between two adjacent unit fractions of the sequence there is an interval.

> > THIS IMPLIES Between all x > 0

> No, *each* (every individual such element) is not the same as *all*
> (the set of such elements).

I have not used the set. I have used all points x that can be chosen and have ℵ₀ unit fractions between themselves and 0.

Regards, WM

William schrieb am Sonntag, 12. Februar 2023 um 18:25:23 UTC+1:
> On Sunday, February 12, 2023 at 12:35:19 PM UTC-4, WM wrote:
> > William schrieb am Sonntag, 12. Februar 2023 um 17:28:50 UTC+1:
> > > On Sunday, February 12, 2023 at 12:16:37 PM UTC-4, WM wrote:
> >
> > > > No. |N_P is a potentially infinite collection, always finite but not fixed
> > > Nope, |N_P is a set. It does not change.
> > What about the unit fractions that cannot be chosen?
> They are reciprocals of "dark natural numbers" which are larger than each element of |N_P. Neither the "dark natural numbers" nor their reciprocals are elements of |R.

The dark unit fractions are points on the positive real line:

They are existing, as set theory claims, between 0 and any point x > 0. And they cannot be outside of the interval (0, 1]. Therefore their reciprocals are real numbers.

Regards, WM

On Sunday, February 12, 2023 at 2:48:45 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 12. Februar 2023 um 18:25:23 UTC+1:
> > On Sunday, February 12, 2023 at 12:35:19 PM UTC-4, WM wrote:
> > > William schrieb am Sonntag, 12. Februar 2023 um 17:28:50 UTC+1:
> > > > On Sunday, February 12, 2023 at 12:16:37 PM UTC-4, WM wrote:
> > >
> > > > > No. |N_P is a potentially infinite collection, always finite but not fixed
> > > > Nope, |N_P is a set. It does not change.
> > > What about the unit fractions that cannot be chosen?
> > They are reciprocals of "dark natural numbers" which are larger than each element of |N_P. Neither the "dark natural numbers" nor their reciprocals are elements of |R.
> The dark unit fractions
The things you call "dark unit factions" are reciprocals of "dark natural numbers" They are not elements of R.

WM was thinking very hard :
> William schrieb am Sonntag, 12. Februar 2023 um 18:25:23 UTC+1:
>> On Sunday, February 12, 2023 at 12:35:19 PM UTC-4, WM wrote:
>>> William schrieb am Sonntag, 12. Februar 2023 um 17:28:50 UTC+1:
>>>> On Sunday, February 12, 2023 at 12:16:37 PM UTC-4, WM wrote:
>>>>> No. |N_P is a potentially infinite collection, always finite but not
>>>>> fixed
>>>> Nope, |N_P is a set. It does not change.
>>> What about the unit fractions that cannot be chosen?
>> They are reciprocals of "dark natural numbers" which are larger than each
>> element of |N_P. Neither the "dark natural numbers" nor their reciprocals
>> are elements of |R.
>
> The dark unit fractions are points on the positive real line:
>
> They are existing, as set theory claims, between 0 and any point x > 0.

Set theory makes no such claim, that is just your misconception of set
theory.

crank Wolfgang Mückenheim, aka WM wrote:
> Python schrieb am Sonntag, 12. Februar 2023 um 15:36:52 UTC+1:
>> crank Wolfgang Mückenheim, aka WM wrote:
>
>>> Do you really claim that there are ℵo different points between 0 and (0, 1]? That will be a good teaser for my next lecture.
>
>>> I would be intersted if anybody shares your opinion. Not even YBM I think will be so fanatic.
>> There is no fanatism there, only sound logic and mathematics. Ben
>> Bacarisse is right, as usual, and you, crank Wolfgang Mückenheim,
>> from Hochschule Augsburg, are wrong, as always.
>
> Sorry

The only think you should be sorry of is that you have molested
and abused students for decade and authored a shitty book, crank
Wolfgang Mückenheim, from Hochschule Augsburg.

William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:

> The things you call "dark unit factions" are reciprocals of "dark natural numbers" They are not elements of R.

For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
Do you agree?
At x = 0 there is no unit fraction between 0 and x.
Do you agree?
That means when the cursor moves from 1 to 0, these unit fractions lie before it all the way on the positive real axis. But suddenly they disappear when the cursor reaches 0.
Do you agree?
Can ℵo points of unit fractions together with the intervals between adjacent unit fractions disappear suddenly in one point?

Regards, WM

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 20:04:42 UTC+1:
> WM was thinking very hard :

> > They are existing, as set theory claims, between 0 and any point x > 0.
> Set theory makes no such claim, that is just your misconception of set
> theory.

Really? Ask the experts.

Regards, WM

On Sunday, February 12, 2023 at 8:28:22 PM UTC+1, WM wrote:
> FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 20:04:42 UTC+1:
> > WM was thinking very hard :
> > >
> > > [Dark unit fractions] are existing, as set theory claims, between 0 and any point x > 0.
> > >
> > Set theory makes no such claim, that is just your misconception of set theory.
> >
> Really? Ask the experts.

Can you quote such an expert, you psychotic asshole full of shit?

I can: "[WM’s] conclusions are based on the sloppiness of his notions, his inability of giving precise definitions, his fundamental misunderstanding of elementary mathematical concepts, and sometimes, as the late Dik Winter remarked [...], on nothing at all." (Franz Lemmermeyer)

On Sunday, February 12, 2023 at 3:26:24 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:
>
> > The things you call "dark unit factions" are reciprocals of "dark natural numbers" They are not elements of R.
> For every point x ∈ (0, 1]

Every point in the segment of the real number line (0,1] corresponds to an element of |R. No reciprocals the things you call "dark natural numbers" are elements of |R.

Fritz Feldhase wrote:

> On Sunday, February 12, 2023 at 8:28:22 PM UTC+1, WM wrote:
>> Really? Ask the experts.
>
> Can you quote such an expert, you psychotic asshole full of shit?

so true indeed. But remember that immediate after the terrorism
*_blowing_up_* the *_gas_pipeline_* in the middle of the sea, the
*_Norway_opened_a_gas_pipeline_* to Poland, but given and *_inaugurated_*
by the *_khazar_danish_prime_minister_*, not the Norway one. Why, not to
make it *_too_obvious_*. So both *_america,_norway,_denmark_and_poland_*
are *_collaborators_* in the *_terrorism_* and the *_war_crime_*. Which
proves once again, that Hitler *_went_to_war_* against *_wrong_enemy_*.
The khazars *_stick_together_*, whereever money, *_fake_money_* and goods
are involved on the planet.

https://www.euractiv.com/wp-content/uploads/sites/2/2022/09/
w_57954926-800x450.jpg

https://www.euractiv.com/section/energy-environment/news/baltic-pipe-gas-
pipeline-opens-connects-norway-and-poland/

*/_How_America_Blew_Up_The_Nord_Stream_Pipeline_/* ｜ Redacted with Natali
and Clayton Morris -[09⧸02⧸2023]
https://%62%69%74%63%68%75%74%65.com/%76%69%64%65%6f/bkuqrojZynOC

This Russian Offensive in Ukraine - *_Scott_Ritter_* - [10⧸02-2023]
https://%62%69%74%63%68%75%74%65.com/%76%69%64%65%6f/tZcSu3uWMB3L

Sorry, this is sci.math. We discuss harebrained math theories here. Not harebrained conspiracy theories.

WM pretended :
> William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:
>
>> The things you call "dark unit factions" are reciprocals of "dark natural
>> numbers" They are not elements of R.
>
> For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
> Do you agree?
> At x = 0 there is no unit fraction between 0 and x.
> Do you agree?
> That means when the cursor moves from 1 to 0, these unit fractions lie before
> it all the way on the positive real axis. But suddenly they disappear when
> the cursor reaches 0. Do you agree? Can ℵo points of unit fractions together
> with the intervals between adjacent unit fractions disappear suddenly in one
> point?

Yes, but it is purely coincidental.

On Sunday, February 12, 2023 at 12:23:39 PM UTC-8, William wrote:
> Sorry, this is sci.math. We discuss harebrained math theories here. Not harebrained conspiracy theories.

Hare-brained, like a brain the size of a rabbit, with its nose all twitchy and eyes all bulging out?

William schrieb am Sonntag, 12. Februar 2023 um 21:09:32 UTC+1:
> On Sunday, February 12, 2023 at 3:26:24 PM UTC-4, WM wrote:
> > William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:
> >
> > > The things you call "dark unit factions" are reciprocals of "dark natural numbers" They are not elements of R.
> > For every point x ∈ (0, 1]
> Every point in the segment of the real number line (0,1] corresponds to an element of |R.

Yes.

> No reciprocals the things you call "dark natural numbers" are elements of |R.

Obviously wrong. Can't you answer the questions I asked you one after the other?

For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
Do you agree?
At x = 0 there is no unit fraction between 0 and x.
Do you agree?
That means when the cursor moves from 1 to 0, these unit fractions lie before it all the way on the positive real axis. But suddenly they disappear when the cursor reaches 0.
Do you agree?
Can ℵo points of unit fractions together with the intervals between adjacent unit fractions disappear suddenly in one point?

Regards, WM

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 21:46:22 UTC+1:
> WM pretended :
> > William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:
> >
> >> The things you call "dark unit factions" are reciprocals of "dark natural
> >> numbers" They are not elements of R.
> >
> > For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
> > Do you agree?
> > At x = 0 there is no unit fraction between 0 and x.
> > Do you agree?
> > That means when the cursor moves from 1 to 0, these unit fractions lie before
> > it all the way on the positive real axis. But suddenly they disappear when
> > the cursor reaches 0. Do you agree? Can ℵo points of unit fractions together
> > with the intervals between adjacent unit fractions disappear suddenly in one
> > point?
> Yes

No. That is impossible.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 12. Februar 2023 um 20:49:56 UTC+1:
> On Sunday, February 12, 2023 at 8:28:22 PM UTC+1, WM wrote:
> > FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 20:04:42 UTC+1:
> > > WM was thinking very hard :
> > > >
> > > > [ℵo unit fractions] are existing, as set theory claims, between 0 and any point x > 0.
> > > >
> > > Set theory makes no such claim, that is just your misconception of set theory.
> > >
> > Really? Ask the experts.
> Can you quote such an expert,

Why should I? We both know it.

Regards, WM

On Sunday, February 12, 2023 at 5:38:37 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 12. Februar 2023 um 21:09:32 UTC+1:
> > On Sunday, February 12, 2023 at 3:26:24 PM UTC-4, WM wrote:
> > > William schrieb am Sonntag, 12. Februar 2023 um 19:57:47 UTC+1:
> > >
> > > > The things you call "dark unit factions" are reciprocals of "dark natural numbers" They are not elements of R.
> > > For every point x ∈ (0, 1]
> > Every point in the segment of the real number line (0,1] corresponds to an element of |R.
> Yes.
> > No reciprocals the things you call "dark natural numbers" are elements of |R.
> Obviously wrong.
Nope, the things you call "dark natural numbers" are larger than each element of |N_P. No element of |R is larger than each element of |N_P (this is just saying that every element of |R is finite).

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Sonntag, 12. Februar 2023 um 17:36:51 UTC+1:
>> We have the WM-startling
>> fact that as x passes smoothly from, say 1 to -1, the number of unit
>> fractions between x and zero goes from infinity to zero in an instant!
>
> Between every x > 0 that can be chosen and 0 there are ℵ₀ unit
> fractions as geometric points (and in addition they have intervals
> between them).
> THIS IMPLIES
> Between all x > 0 that can be chosen and 0 there are ℵ₀ unit
> fractions as geometric points (and in addition they have intervals
> between them). They cannot go in an instant!
>
> HOWEVER
> Between all x > 0 and 0 there is nothing.
>
> Hence many points cannot be chosen. They are dark.

Not bad, but I think a few students might still see though this waffle.
I thought your original misuse of "distance" was a better way to create
confusion, but if you think this flipping from "every" to "all" in one
sense and then to "all" in the other sense is your best bet, then go for
it!

--
Ben.

WM <askasker48@gmail.com> writes:

> Ben Bacarisse schrieb am Sonntag, 12. Februar 2023 um 02:12:13 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>> >> >> Both proper mathematics and WMaths agree that every FISON has infinitely
>> >> >> many successors[1] and that a set of FISONS is not a natnumber and
>> >> >> therefore has no natnumber successor.
>> >> >
>> >> > Instead of the set consider all FISONs.
>> >> I considered and addressed what you said. We both agree on these two
>> >> points
>> >
>> > Not at all! The set of FISONs is a FISON in classical
>> > mathematics. This was changed only by Cantor and Bolzano.
>
>> | ... he knows that 0 is the GLB of U (where U is {1/n} with n in N⁺)
>
> Of course.
>
>> | so that {u | u ∈ U and u <= x} is empty for x <= 0 and not finite
>> | when x > 0 so I doubt he disputes any matter of fact here.
>
> Wrong.

Oh! An actual disagreement about something that is not too vague to pin
down. Which of the two matters of fact are you saying is/are wrong?

(a) {u | u ∈ U and u <= x} is empty for x <= 0
(b) {u | u ∈ U and u <= x} is not finite when x > 0

--
Ben.

William schrieb am Sonntag, 12. Februar 2023 um 23:36:07 UTC+1:
> On Sunday, February 12, 2023 at 5:38:37 PM UTC-4, WM wrote:

> > > No reciprocals the things you call "dark natural numbers" are elements of |R.
> > Obviously wrong.
> Nope, the things you call "dark natural numbers" are larger than each element of |N_P. No element of |R is larger than each element of |N_P (this is just saying that every element of |R is finite).

Can't you refrain from stating unfounded claims? Please answer this step by step:

For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
Do you agree?
At x = 0 there is no unit fraction between 0 and x.
Do you agree?
That means when the cursor moves from 1 to 0, these unit fractions lie before it all the way on the positive real axis. But suddenly they disappear when the cursor reaches 0.
Do you agree?
Can ℵo points of unit fractions together with the intervals between adjacent unit fractions disappear suddenly in one point?

Regards, WM