Simply answer by yes or no:
1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming Cantor's metric.)
2) Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?

Regards, WM

WM submitted this idea :
> Simply answer by yes or no:
> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming
> Cantor's metric.)

Of course, countably many unit fractions are always going to be a
smaller set than the uncountably many real numbers.

> 2) Can the intersection of the infinite terms of an inclusion-monotonic
> sequence be empty?

Of course, especially when the sets don't *all* have at least one of
*the same* element but only the same size as a set.

You still don't get it, do you professor. :)

On Tuesday, September 12, 2023 at 2:44:08 PM UTC+2, WM wrote:

> 2) Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?

Yes. A trivial fact.

Consider the set sequence (E_n) defined with E_n = {n, n+1, n+2, ...} (for all n e IN).

This sequence is inclusion-monotonic (An e IN: E_(n+1) c E_n) and for all n e IN: E_n is infinite.

INTERSECTION_(n e IN) E_n = { }.

Fritz Feldhase schrieb am Dienstag, 12. September 2023 um 17:27:07 UTC+2:
> On Tuesday, September 12, 2023 at 2:44:08 PM UTC+2, WM wrote:
>
> > 2) Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?
> Yes. A trivial fact.
>
> Consider the set sequence (E_n) defined with E_n = {n, n+1, n+2, ...} (for all n e IN).
>
> This sequence is inclusion-monotonic (An e IN: E_(n+1) c E_n) and for all n e IN: E_n is infinite.
>
> INTERSECTION_(n e IN) E_n = { }.

No. Why do you believe in this mistake? By the faulty argument that all natnumbers can be removed by removing them one by one. That is wrong. To remove all we must remove the set collectively. But that does not happen with endsegments: ∀k ∈ ℕ: E(k+1) = E(k) \ {k}.

Regards, WM

FromTheRafters schrieb am Dienstag, 12. September 2023 um 16:43:46 UTC+2:
> WM submitted this idea :
> > Simply answer by yes or no:
> > 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming
> > Cantor's metric.)
> Of course, countably many unit fractions are always going to be a
> smaller set than the uncountably many real numbers.

Can you represent two points without a gap between them?

> > 2) Can the intersection of the infinite terms of an inclusion-monotonic
> > sequence be empty?
> Of course,

Never. All infinite sets contain one and the same infinite subset. Inclusion monotony.

Regards, WM

On Tuesday, September 12, 2023 at 10:43:45 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 12. September 2023 um 17:27:07 UTC+2:
> > On Tuesday, September 12, 2023 at 2:44:08 PM UTC+2, WM wrote:
> > >
> > > Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?
> > >
> > Yes. A trivial fact.
> >
> > Consider the set sequence (E_n) defined with E_n = {n, n+1, n+2, ...} (for all n e IN).
> >
> > This sequence is inclusion-monotonic (An e IN: E_(n+1) c E_n) and for all n e IN: E_n is infinite.
> >
> > INTERSECTION_(n e IN) E_n = { }.
> >
> No.

Yes, und nun hau endlich ab Du hirnloses, krankes Arschloch. Lass Dich endlich mal behandeln!

You are a disgusting pest.

EOD

On 9/12/2023 8:44 AM, WM wrote:

> 2) Can the intersection of the infinite terms
> of an inclusion-monotonic sequence be empty?

Yes.
Consider
infinite inclusion-monotonic ⟨A₀ A₁ A₂ …⟩
Inclusion-monotonic Aⱼ ⊃≠ Aⱼ₊₁
Infinite |⟨Aⱼ Aⱼ₊₁ Aⱼ₊₂ …⟩| = ℵ₀

Define
difference sequence ⟨D₀ D₁ D₂ …⟩
Dⱼ = Aⱼ\Aⱼ₊₁
|Dⱼ| ≥ 1
j≠k ⇒ Dⱼ∩Dₖ = ∅

Define
end-union sequence ⟨U₀ U₁ U₂ …⟩
Uⱼ = ⋃⟨Dⱼ Dⱼ₊₁ Dⱼ₊₂ …⟩
|Uⱼ| ≥ |⟨Dⱼ Dⱼ₊₁ Dⱼ₊₂ …⟩| = ℵ₀
|Uⱼ| ≥ ℵ₀
Uⱼ ⊃≠ Uⱼ₊₁

⋂⟨U₀ U₁ U₂ …⟩ = ∅

| Assume otherwise.
| Assume ⋂⟨U₀ U₁ U₂ …⟩ ∋ d
| | U₀ ∋ d
| U₀ = ⋃⟨D₀ D₁ D₂ …⟩
| ∃Dⱼ ∋ d
| ∀Dₖ ∈ ⟨Dⱼ₊₁ Dⱼ₊₂ …⟩: Dₖ∩Dⱼ = ∅
| ∀Dₖ ∈ ⟨Dⱼ₊₁ Dⱼ₊₂ …⟩: Dₖ ∌ d
| U⟨Dⱼ₊₁ Dⱼ₊₂ …⟩ ∌ d
| Uⱼ₊₁ ∌ d
| ⋂⟨U₀ U₁ U₂ …⟩ ∌ d
| Contradiction.

Therefore,
this infinite sequence of infinite sets
|⟨U₀ U₁ U₂ …⟩| = ℵ₀
|Uⱼ| ≥ ℵ₀
has an empty intersection.
|⋂⟨U₀ U₁ U₂ …⟩| = 0

WM expressed precisely :
> FromTheRafters schrieb am Dienstag, 12. September 2023 um 16:43:46 UTC+2:
>> WM submitted this idea :
>>> Simply answer by yes or no:
>>> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming
>>> Cantor's metric.)
>> Of course, countably many unit fractions are always going to be a
>> smaller set than the uncountably many real numbers.
>
> Can you represent two points without a gap between them?

There are no gaps between real numbers. Isolated points, in a discrete
set, have gaps in the sense that some reals would be placed in between
them when embedded in the reals.

>>> 2) Can the intersection of the infinite terms of an inclusion-monotonic
>>> sequence be empty?
>> Of course,
>
> Never. All infinite sets contain one and the same infinite subset. Inclusion
> monotony.

Because you say so?

On Wednesday, September 13, 2023 at 1:18:02 AM UTC+3, FromTheRafters wrote:
> WM expressed precisely :
> > FromTheRafters schrieb am Dienstag, 12. September 2023 um 16:43:46 UTC+2:
> >> WM submitted this idea :
> >>> Simply answer by yes or no:
> >>> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming
> >>> Cantor's metric.)
> >> Of course, countably many unit fractions are always going to be a
> >> smaller set than the uncountably many real numbers.
> >
> > Can you represent two points without a gap between them?
> There are no gaps between real numbers. Isolated points, in a discrete
> set, have gaps in the sense that some reals would be placed in between
> them when embedded in the reals.
> >>> 2) Can the intersection of the infinite terms of an inclusion-monotonic
> >>> sequence be empty?
> >> Of course,
> >
> > Never. All infinite sets contain one and the same infinite subset. Inclusion
> > monotony.
> Because you say so?

Talking interms of a language that academic mathematicians may understand, in order to describe the issue of a set or numbers; I only see the following:

Between two given distinct "constructible " numbers, no matter however close to each other YOU may think, or no matter however far from each others You may think, then Between them there are infinitely many more & only "constructible "numbers between them, where no other types of real numbers ever exist between them

..Where if you think otherwise, then simply prove your insight 😉!

Where also it is absolutely impossible to prove your insight, which are only & strictly mere & false symbolic beileafs, FOR SURE

Bassam karzeddin

On Wednesday, September 13, 2023 at 4:16:42 AM UTC+3, bassam karzeddin wrote:
> On Wednesday, September 13, 2023 at 1:18:02 AM UTC+3, FromTheRafters wrote:
> > WM expressed precisely :
> > > FromTheRafters schrieb am Dienstag, 12. September 2023 um 16:43:46 UTC+2:
> > >> WM submitted this idea :
> > >>> Simply answer by yes or no:
> > >>> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming
> > >>> Cantor's metric.)
> > >> Of course, countably many unit fractions are always going to be a
> > >> smaller set than the uncountably many real numbers.
> > >
> > > Can you represent two points without a gap between them?
> > There are no gaps between real numbers. Isolated points, in a discrete
> > set, have gaps in the sense that some reals would be placed in between
> > them when embedded in the reals.
> > >>> 2) Can the intersection of the infinite terms of an inclusion-monotonic
> > >>> sequence be empty?
> > >> Of course,
> > >
> > > Never. All infinite sets contain one and the same infinite subset. Inclusion
> > > monotony.
> > Because you say so?
> Talking interms of a language that academic mathematicians may understand, in order to describe the issue of a set or numbers; I only see the following:
>
> Between two given distinct "constructible " numbers, no matter however close to each other YOU may think, or no matter however far from each others You may think, then Between them there are infinitely many more & only "constructible "numbers between them, where no other types of real numbers ever exist between them
>
> .Where if you think otherwise, then simply prove your insight 😉!
>
> Where also it is absolutely impossible to prove your insight, which are only & strictly mere & false symbolic beileafs, FOR SURE
>
> Bassam karzeddin

..By the way!

Do the academic mainstreams mathematicians & alike well-understand what does a "constructible " number mean exactly in their mathematics? Wonder!

BKK

FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
> WM expressed precisely :
> > All infinite sets contain one and the same infinite subset. Inclusion
> > monotony.
> Because you say so?

Because that is the definition of inclusion monotony for infinite sets.

Regards, WM

bassam karzeddin schrieb am Mittwoch, 13. September 2023 um 03:16:42 UTC+2:

> Between two given distinct "constructible " numbers, no matter however close to each other YOU may think, or no matter however far from each others You may think, then Between them there are infinitely many more & only "constructible "numbers between them, where no other types of real numbers ever exist between them
>
> .Where if you think otherwise, then simply prove your insight 😉!

That is impossible. Numbers that are not constructible do only exist because Cantor's actual infinity is assumed. Without that only potential infinity exists and you are right.

Regards, WM

On Wednesday, 13 September 2023 at 12:15:19 UTC-3, WM wrote:
> FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
> > WM expressed precisely :
> > > All infinite sets contain one and the same infinite subset. Inclusion
> > > monotony.
> > Because you say so?
> Because that is the definition of inclusion monotony for infinite sets.

Muckenheim, you are just too dumb and too stupid. You can claim to know what inclusion monotony means until the cows come home, but you will never understand it.

Gus Gassmann schrieb am Mittwoch, 13. September 2023 um 18:42:46 UTC+2:
> On Wednesday, 13 September 2023 at 12:15:19 UTC-3, WM wrote:
> > FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
> > > WM expressed precisely :
> > > > All infinite sets contain one and the same infinite subset. Inclusion
> > > > monotony.
> > > Because you say so?
> > Because that is the definition of inclusion monotony for infinite sets.
> You can claim to know what inclusion monotony means

It is simple: The sets only shrink but do not grow. Therefore an infinite contents is the same in all infinite sets. And a common contents is in all non-empty sets.

Regards, WM

WM laid this down on his screen :
> bassam karzeddin schrieb am Mittwoch, 13. September 2023 um 03:16:42 UTC+2:
>
>> Between two given distinct "constructible " numbers, no matter however close
>> to each other YOU may think, or no matter however far from each others You
>> may think, then Between them there are infinitely many more & only
>> "constructible "numbers between them, where no other types of real numbers
>> ever exist between them
>>
>> .Where if you think otherwise, then simply prove your insight ?!
>
> That is impossible. Numbers that are not constructible do only exist because
> Cantor's actual infinity is assumed.

???

> Without that only potential infinity exists and you are right.

Wrong, real numbers are the set we created to deal with the fact that
not all numbers are geometrically constructible or algebraic.

WM explained on 9/13/2023 :
> Gus Gassmann schrieb am Mittwoch, 13. September 2023 um 18:42:46 UTC+2:
>> On Wednesday, 13 September 2023 at 12:15:19 UTC-3, WM wrote:
>>> FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
>>>> WM expressed precisely :
>>>>> All infinite sets contain one and the same infinite subset. Inclusion
>>>>> monotony.
>>>> Because you say so?
>>> Because that is the definition of inclusion monotony for infinite sets.
>> You can claim to know what inclusion monotony means
>
> It is simple: The sets only shrink but do not grow. Therefore an infinite
> contents is the same in all infinite sets. And a common contents is in all
> non-empty sets.

Yep, you just proved Gus Gassmann's theory correct.

On Wednesday, 13 September 2023 at 14:26:05 UTC-3, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 13. September 2023 um 18:42:46 UTC+2:
> > On Wednesday, 13 September 2023 at 12:15:19 UTC-3, WM wrote:
> > > FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
> > > > WM expressed precisely :
> > > > > All infinite sets contain one and the same infinite subset. Inclusion
> > > > > monotony.
> > > > Because you say so?
> > > Because that is the definition of inclusion monotony for infinite sets.
> > You can claim to know what inclusion monotony means
> It is simple: The sets only shrink but do not grow. Therefore an infinite contents is the same in all infinite sets.

Simply not so. And all your protestations to the contrary, you are just too dumb and too stupid.

Gus Gassmann schrieb am Mittwoch, 13. September 2023 um 19:44:11 UTC+2:
> On Wednesday, 13 September 2023 at 14:26:05 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Mittwoch, 13. September 2023 um 18:42:46 UTC+2:
> > > On Wednesday, 13 September 2023 at 12:15:19 UTC-3, WM wrote:
> > > > FromTheRafters schrieb am Mittwoch, 13. September 2023 um 00:18:02 UTC+2:
> > > > > WM expressed precisely :
> > > > > > All infinite sets contain one and the same infinite subset. Inclusion
> > > > > > monotony.
> > > > > Because you say so?
> > > > Because that is the definition of inclusion monotony for infinite sets.
> > > You can claim to know what inclusion monotony means
> > It is simple: The sets only shrink but do not grow. Therefore an infinite contents is the same in all infinite sets.
> Simply not so.

Go, inform you about iclusion-monotony. For all infinite endsegments which can be determined, we have ∀k ∈ ℕ_det: |∩{E(1), E(2), ..., E(k)}| = ℵ₀. If there were no others, then this would be true for all endsegments.

But you know also ∀k ∈ ℕ: E(k+1) = E(k) \ {k} and you extrapolate it to all endsegments. This yields the empty intersection. But it yields an empty endsegment as well.

Regards, WM

On Tuesday, September 12, 2023 at 3:44:08 PM UTC+3, WM wrote:
> Simply answer by yes or no:
> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming Cantor's metric.)
> 2) Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?
>
> Regards, WM

There are also many more tools that proves the unbelievable global stupidity among the vast majorities of mainstream academic proffessional mathematicians which are too easy even for mid-school students to understand
The easiest tool I think is the use of the decimal notation only when expressing their beliefs about their top-most worshiped & hollyest numbers like Pi, Cubrt for example, where if you request their alleged top-most genius & well-known mathematicians to express those numbers but without using the decimal notation only for one time in their meaningless lives, they would certainly reject to answer (as if the decimal notation is a magical tool that can generate infinitely many real numbers so cheaply as they like

But the decimal notation is only a kind of division operation but they don't want to comprehend this too simple truth 🤔!

However, many more tools were freely & publically published in my remaining & unstolen posts for sure

And mathematics become as an art of how humans can be so stupids up to this level & openly & globally under the sun light 🤔!

Bassam karzeddin

On Wednesday, September 13, 2023 at 11:54:18 PM UTC+2, bassam karzeddin wrote:

> And mathematics become as an art of how humans can be so stupids up to this level & openly & globally under the sun light 🤔!

For sure!

On Thursday, September 14, 2023 at 1:00:44 AM UTC+3, Fritz Feldhase wrote:
> On Wednesday, September 13, 2023 at 11:54:18 PM UTC+2, bassam karzeddin wrote:
>
> > And mathematics become as an art of how humans can be so stupids up to this level & openly & globally under the sun light 🤔!
> For sure!

Dare you express your Pi but without using the decimal notation!?
Hint: (3.14 = 314/100), Right ✅️?

Go a head & continue if you have the Guts & tell us what would you arrive at finally?

I know this is too embarrassing for ALL academics & not only for the poorest minds of mathematicians, but also for all theoretical academics as (logicians, philosophers, physicians,..., etc) & of course FOR SURE

BKK

Le 14/09/2023 à 02:40, bASSam karzeddin a écrit :
> On Thursday, September 14, 2023 at 1:00:44 AM UTC+3, Fritz Feldhase wrote:
>> On Wednesday, September 13, 2023 at 11:54:18 PM UTC+2, bassam karzeddin wrote:
>>
>>> And mathematics become as an art of how humans can be so stupids up to this level & openly & globally under the sun light 🤔!
>> For sure!
>
> Dare you express your Pi but without using the decimal notation!?
> Hint: (3.14 = 314/100), Right ✅️?
>
> Go a head & continue if you have the Guts & tell us what would you arrive at finally?
>
> I know this is too embarrassing for ALL academics & not only for the poorest minds of mathematicians, but also for all theoretical academics as (logicians, philosophers, physicians,..., etc) & of course FOR SURE

Well. No. Not at all. You fail to get it, so what?

You are an idiot, you'll stay an idiot, you'll die as such. So what?

Does you family knows, Mr Karzeddin?

On Thursday, September 14, 2023 at 3:47:22 AM UTC+3, Python wrote:
> Le 14/09/2023 à 02:40, bASSam karzeddin a écrit :
> > On Thursday, September 14, 2023 at 1:00:44 AM UTC+3, Fritz Feldhase wrote:
> >> On Wednesday, September 13, 2023 at 11:54:18 PM UTC+2, bassam karzeddin wrote:
> >>
> >>> And mathematics become as an art of how humans can be so stupids up to this level & openly & globally under the sun light 🤔!
> >> For sure!
> >
> > Dare you express your Pi but without using the decimal notation!?
> > Hint: (3.14 = 314/100), Right ✅️?
> >
> > Go a head & continue if you have the Guts & tell us what would you arrive at finally?
> >
> > I know this is too embarrassing for ALL academics & not only for the poorest minds of mathematicians, but also for all theoretical academics as (logicians, philosophers, physicians,..., etc) & of course FOR SURE
> Well. No. Not at all. You fail to get it, so what?
>
> You are an idiot, you'll stay an idiot, you'll die as such. So what?
>
> Does you family knows, Mr Karzeddin?

I do understand your unlimited frustration, but the truth is actually so ugly Mr. Python

But slowly, slowly you must get used to it since there isn't any other way to hide the ugly truth by by a spider threads any more, where also, I did teach you many other ways earlier about Pi being actually a variable constructible number (which is never any constant number) & solely a property of regular existing polygons, where also, your Pi in mind for the so-called 🔵 circle doesn’t exist nor it is a number FOR MORE THAN SURE

Should I keep repeating ALL my previous public published proofs that are irrefuitable for each blind dogmatic believer in this utter century? Wonder!

Pick up fast the truth instead of fighting aimlessly about it & do spread it globally as well

Bassam karzeddin 🔊

On Tuesday, September 12, 2023 at 3:44:08 PM UTC+3, WM wrote:
> Simply answer by yes or no:
> 1) Can ℵ₀ unit fractions occupy less than ℵ₀*2^ℵ₀ points? (When assuming Cantor's metric.)
> 2) Can the intersection of the infinite terms of an inclusion-monotonic sequence be empty?
>
> Regards, WM

Actually, we must invent tools to dedect the clever mathematicians since stupidity of academic mainstreams proffessional mathematicians is already well-proven not only by modest self but also by few more distinguished ginious people as well

Which make us so suspicious about the standard IO tests usually they adopt 🤔!

BKK

On Wednesday, 13 September 2023 at 17:33:38 UTC-3, WM wrote:
[...]
> Go, inform you about iclusion-monotony.

Right back atcha. But wait! You are just too dumb and too stupid.