On Tuesday, December 7, 2021 at 12:46:21 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 7. Dezember 2021 um 16:03:20 UTC+1:
> > On Tuesday, December 7, 2021 at 7:57:36 AM UTC-4, WM wrote:
> >
> > You note
> > > ∀k ∈ [ℕ_P]: ∩{E(1), E(2), ..., E(k)} [<snip>]| = ℵ₀
> > > This shows that the collection of collected infinite endsegments with infinite intersection is infinite.
> > This collection is also a set with cardinality aleph_0. Potentially infinite set is nonsense
> This [set] cannot have cardinality ℵ₀ because all endsegments have ℵ₀ numbers as content.

Indeed, the elements of the set are sets with aleph_0 elements. So what? The set has cardinality aleph_0. As usual you confuse the elements of the set and the set.

> Therefore there cannot exist ℵ₀ indexes smaller than the contents.

There is no *the* contents. Each element has a different set of contents

Each element of the set of natural numbers is finite. The set of natural numbers has cardinality aleph_0
--
William Hughes

tisdag 7 december 2021 kl. 13:01:14 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 7. Dezember 2021 um 06:36:57 UTC+1:
> > måndag 6 december 2021 kl. 11:45:31 UTC+1 skrev WM:
> > > zelos...@gmail.com schrieb am Montag, 6. Dezember 2021 um 06:23:22 UTC+1:
> > > > fredag 3 december 2021 kl. 15:08:48 UTC+1 skrev WM:
> > >
> > > > > Collecting natural numbers yields this set. Collecting only definable natural numbers yields finite sets with ℵo successors before ω. You will agree that this cannot be changed.
> > > > >
> > > > > Collecting all natnumbers however yields a set with no successors before ω.
> > > > >
> > > > > The (more or less sudden) change cannot be observed.
> > > > > Anyhow, this shows that something here cannot be obeserved.
> > > > > Do you agree that something here cannot be observed?
> > > > >
> > > > Your set of "definable natural numbers" is not finite
> > > It is potentially infinite. But it cannot be an ℵo-set, because it is followed by an ℵo-set.
> > >
> > no, it is INFINITE.
>
> Potentially infinite, to be precise.
> Every endsegment satifies the rule
> If
> |∩{E(1), E(2), ..., E(n)}| = ℵ₀
> then
> |∩{E(1), E(2), ..., E(n+1)}| = ℵ₀
> because
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> This shows that the collection of collected infinite endsegments with infinite intersection is infinite. But it is not the ℵ₀-set of all endsegments because
> ∩{E(1), E(2), ... } = { }.
> "Every finite set of endsegments" is the name of a potentially infinite collection of infinite endsegments. Do you agree? If not, what is your interpretatition?
>
> Regards, WM

There is no such thing as "potential infinite"

tisdag 7 december 2021 kl. 12:57:36 UTC+1 skrev WM:
> William schrieb am Dienstag, 7. Dezember 2021 um 02:05:31 UTC+1:
> > On Monday, December 6, 2021 at 5:43:11 AM UTC-5, WM wrote:
> > > potentially infinite.
> >
> >
> > Potentially infinite set is nonsense.
> Every endsegment satifies the rule
> If
> |∩{E(1), E(2), ..., E(n)}| = ℵ₀
> then
> |∩{E(1), E(2), ..., E(n+1)}| = ℵ₀
> because
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> This shows that the collection of collected infinite endsegments with infinite intersection is infinite. Do you agree?
>
> "Every finite set of endsegments" is the name of a potentially infinite collection of infinite endsegments. Do you agree? If not, what is your interpretatition?
>
> Regards, WM
False

It does NOT show that. It shows only that FINITE collections has an infinite intersection.

tisdag 7 december 2021 kl. 13:18:14 UTC+1 skrev WM:
> Fritz Feldhase schrieb am Dienstag, 7. Dezember 2021 um 13:16:12 UTC+1:
> > On Tuesday, December 7, 2021 at 1:01:14 PM UTC+1, WM wrote:
> > > zelos...@gmail.com schrieb am Dienstag, 7. Dezember 2021 um 06:36:57 UTC+1:
> > > > måndag 6 december 2021 kl. 11:45:31 UTC+1 skrev WM:
> > > > > zelos...@gmail.com schrieb am Montag, 6. Dezember 2021 um 06:23:22 UTC+1:
> > > > > >
> > > > > > Your set of "definable natural numbers" is not finite
> > > > > >
> > > > > It is potentially infinite. But <bla>
> > > > >
> > > > no, it is INFINITE.
> > > >
> > > Potentially infinite, to <bla>
> >
> > In set theory sets are either finite or infinite (i. e. not finite).
> That is the reason why it is completely useless nonsense with no relation to logic.
>
> Regards, WM

Set Theory works perfectly with logic, the issue is you are incapable of DOING logic

On 12/7/2021 10:41 AM, Gus Gassmann wrote:
> On Tuesday, 7 December 2021 at 12:30:16 UTC-4, Fritz Feldhase wrote:
>> On Tuesday, December 7, 2021 at 2:54:02 PM UTC+1, WM wrote:
>>
>>> Look above. There you have a potentiall infinite set or better collection.
>> Well, I guess one has to look into your asshole to find such a "collection" of shit.
>
> So you're a proctologist now? You'll probably find all those dark numbers there, too. WM is so anal retentive, he hasn't ever released a single one of them.
>

AHAAHHAH! ROFL!!!!

On 12/5/2021 3:50 PM, FromTheRafters wrote:
> Chris M. Thomasson was thinking very hard :
>> On 12/5/2021 4:47 AM, FromTheRafters wrote:
>>> It happens that Chris M. Thomasson formulated :
>>>> On 12/5/2021 2:06 AM, WM wrote:
>>>>> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>>>>>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>>>>>> Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 19:33:36
>>>>>>> UTC+1:
>>>>>>>>
>>>>>>>> The set of all natural numbers which have ℵ₀ successors is just
>>>>>>>> the set of all natural numbers, SINCE each and every natural
>>>>>>>> number has ℵ₀ successors.
>>>>>>>>
>>>>>>> That is wrong.
>>>>>> No, this is a trivial truth.
>>>>>
>>>>> Each and every definable natural number has ℵo successors. But all
>>>>> natural numbers together are the set of all natural numbers. It has
>>>>> no successors before ω.
>>>>>
>>>>> 1) The union of FISONs is same as the union of all definable
>>>>> natnumbers: ℕ_def.
>>>> [...]
>>>>
>>>> Any natural number is able to be defined? No?
>>>
>>> Every natural number is already defined.
>>
>> Touche! :^D
>
> But things are different in muckymath. The first two axioms seem to be
> the "Axiom Schema of Because I said so" and the "Axiom of Repetition".

Shit happens... Well, damn.

On 12/7/2021 9:19 AM, Serg io wrote:
> On 12/7/2021 10:46 AM, WM wrote:
>> William schrieb am Dienstag, 7. Dezember 2021 um 16:03:20 UTC+1:
>>> On Tuesday, December 7, 2021 at 7:57:36 AM UTC-4, WM wrote:
>>>
>>> You note
>>>> ∀k ∈ [ℕ_P]: ∩{E(1), E(2), ..., E(k)} [<snip>]| = ℵ₀
>>>> This shows that the collection of collected infinite endsegments
>>>> with infinite intersection is infinite.
>>> This collection is also a set with cardinality aleph_0. Potentially
>>> infinite set is nonsense
>>
>> This collection cannot have cardinality ℵ₀ because all endsegments
>> have ℵ₀ numbers as content. Therefore there cannot exist ℵ₀ indexes
>> smaller than the contents.
>
> LAMO!  go buy another ℵ₀ natural numbers off eBay, only $2.31 Euro

The book:

https://youtu.be/rVtHrgdcvZA

;^)

Fritz Feldhase schrieb am Dienstag, 7. Dezember 2021 um 18:06:04 UTC+1:
> On Tuesday, December 7, 2021 at 6:00:59 PM UTC+1, WM wrote:
>
> > Exercise: Prove that the set E of endsegments E(n) with |E(n)| = ℵo cannot have |E| = ℵo.
> What a silly "exercise".

Very important for young students who have not yet comprehended the mistakes of set theory. This exercise shows once and for all that set theory is smashed to pieces by the fact that never two consecutive ℵo-sets can exist in the natural order of ℕ.

Regards, WM

Gus Gassmann schrieb am Dienstag, 7. Dezember 2021 um 19:50:37 UTC+1:
> On Tuesday, 7 December 2021 at 12:52:35 UTC-4, WM wrote:

> > But it is not a finite subset because it has not upper end. It is not an infinite subset, because almost all endsegments are following.
> Each end segment is infinite, but a finite set of end segments is just that: A finite set. There are, of course, (countably) infinitely many such sets.

That is called potential infinity. The set of endsegments with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ is potentially infinite. However the sets which have an empty intersection |∩{E(1), E(2), ...}| = 0, like the set of all endsegments, are actually infinite, that is much more than all finite sets.

> > Obviously it exists. All endsegments have an empty intersection. The potentially infinite collection of infinite endsegments has an infinite intersection.
> The intersection over all possible finite sets of end segments is just as empty as the intersection over the set of all end segments.

That is wrong. Try to understand logic, in particular the universal quantifier, that is the turned A:
∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀

> Your hallucinations are getting weirder and more troublesome by the day.

Chuckle. I use logic.

Regards, WM

William schrieb am Dienstag, 7. Dezember 2021 um 20:26:48 UTC+1:
> On Tuesday, December 7, 2021 at 12:46:21 PM UTC-4, WM wrote:

> > This [set] cannot have cardinality ℵ₀ because all endsegments have ℵ₀ numbers as content.
>
> Indeed, the elements of the set are sets with aleph_0 elements. So what? The set has cardinality aleph_0. As usual you confuse the elements of the set and the set.

The sets catisfying ∀k ∈ ℕ: ∩{E(1), E(2), ...., E(k)} = E(k) /\ |E(k)| = ℵ₀ are less than ℵ₀ because there are not two consecutive ℵ₀-sets in
> > Therefore there cannot exist ℵ₀ indexes smaller than the contents.
> There is no *the* contents. Each element has a different set of contents
>
> Each element of the set of natural numbers is finite. The set of natural numbers has cardinality aleph_0
> --
> William Hughes

WM brought next idea :
>> (1) By ℵo-set you presumably mean a set of cardinality ℵo.
>
>
> In any case we can exclude that two consecutive ℵo-sets cannot exist in the
> natural order of ℕ.

Then stop (lying) trying to pretend that they do. Endsegments are taken
as infinite sets and your sequence of FISONs as sets are simply
infinitely many finite individual sets. "THERE IS NO POTENTIAL
INFINITY" -- the term was made up to appease the matheologians and
philosophers who insisted that physical reality must play a role in
pure mathematics.

William schrieb am Dienstag, 7. Dezember 2021 um 20:26:48 UTC+1:
> On Tuesday, December 7, 2021 at 12:46:21 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 7. Dezember 2021 um 16:03:20 UTC+1:
> > > On Tuesday, December 7, 2021 at 7:57:36 AM UTC-4, WM wrote:
> > >
> > > You note
> > > > ∀k ∈ [ℕ_P]: ∩{E(1), E(2), ..., E(k)} [<snip>]| = ℵ₀
> > > > This shows that the collection of collected infinite endsegments with infinite intersection is infinite.
> > > This collection is also a set with cardinality aleph_0. Potentially infinite set is nonsense
> > This [set] cannot have cardinality ℵ₀ because all endsegments have ℵ₀ numbers as content.
>
> Indeed, the elements of the set are sets with aleph_0 elements. So what? The set has cardinality aleph_0. As usual you confuse the elements of the set and the set.

The sets catisfying ∀k ∈ ℕ: ∩{E(1), E(2), ...., E(k)} = E(k) /\ |E(k)| = ℵ₀ are less than ℵ₀ because there are not two consecutive ℵ₀-sets possible in ℕ.

> > Therefore there cannot exist ℵ₀ indexes smaller than the contents.
> There is no *the* contents.

What shall this nonsense prove? ℵ₀-sets can differ by many elements whithout changing their property of being ℵ₀-set-sets.

> Each element has a different set of contents

Each element has an infinite contents. Therefore it is an ℵ₀-set. That is enough to exclude an infinite sets of indices of these endsegments.
>
> Each element of the set of natural numbers is finite. The set of natural numbers has cardinality aleph_0

The set of definable numbers is potentially infinite. By endsegments you see the difference between pot and act: The set of endsegments with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ is infinite but less than the sets which have an empty intersection |∩{E(1), E(2), ...}| = 0, like the set of all endsegments. They are actually infinite. That is much more than all finite sets.

Regards, WM

zelos...@gmail.com schrieb am Mittwoch, 8. Dezember 2021 um 06:39:11 UTC+1:

> There is no such thing as "potential infinite"

You are wrong. By endsegments you see the difference between pot and act: The set of endsegments with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ is infinite but less than the sets which have an empty intersection |∩{E(1), E(2), ...}| = 0, like the set of all endsegments. They are actually infinite. That is much more than all finite sets.

Regards, WM

zelos...@gmail.com schrieb am Mittwoch, 8. Dezember 2021 um 06:39:46 UTC+1:

> It does NOT show that. It shows only that FINITE collections has an infinite intersection.

Finite without a largest set. That is potential infinity. Or can you find a largest set of endsegements that satisfies ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀?

Regards, WM

FromTheRafters schrieb am Mittwoch, 8. Dezember 2021 um 11:39:19 UTC+1:
> WM brought next idea :
> >> (1) By ℵo-set you presumably mean a set of cardinality ℵo.
> >
> > In any case we can exclude that two consecutive ℵo-sets cannot exist in the
> > natural order of ℕ.

> Endsegments are taken
> as infinite sets and your sequence of FISONs as sets are simply
> infinitely many finite individual sets. "THERE IS NO POTENTIAL
> INFINITY"

Then name the last FISON. The first is {1}. If there is no last, then the sequence of FISONs is potentially infinite. That is so be definition. Shouting will not change this fact.

> the term

was dropped in order to deceive newbies and others who are too stupid to recognize the fraud.

> who insisted that physical reality must play a role in
pure mathematics.

Potential infinity has nothing to do wit reality.

Regards, WM

On Wednesday, 8 December 2021 at 06:16:10 UTC-4, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 7. Dezember 2021 um 18:06:04 UTC+1:
> > On Tuesday, December 7, 2021 at 6:00:59 PM UTC+1, WM wrote:
> >
> > > Exercise: Prove that the set E of endsegments E(n) with |E(n)| = ℵo cannot have |E| = ℵo.
> > What a silly "exercise".
> Very important for young students who have not yet comprehended the mistakes of set theory. This exercise shows once and for all that set theory is smashed to pieces by the fact that never two consecutive ℵo-sets can exist in the natural order of ℕ.

Oh please. Spare us this deranged bullshit, already. Every n ∈ ℕ is finite, and the set E(n) = {n, n+1, n+2, ...} has cardinality ℵo. So the set E is isomorphic to ℕ and hence has cardinality ℵo also. You think by moving from end segments to sets of end segments you have somehow raised the level of sophistication, but all you manage time and again is to deceive and delude yourself. What is next? Sets of sets of sets of sets of end segments? Give us all a break and peddle your bullshit elsewhere.

On Wednesday, 8 December 2021 at 06:30:38 UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 7. Dezember 2021 um 19:50:37 UTC+1:
[...]
> > Your hallucinations are getting weirder and more troublesome by the day.
> Chuckle. I use logic.

That, of course, is the most pernicious of your hallucinations.

On Wednesday, 8 December 2021 at 06:42:17 UTC-4, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 8. Dezember 2021 um 06:39:11 UTC+1:
>
> > There is no such thing as "potential infinite"
> You are wrong. By endsegments you see the difference between pot and act: The set of endsegments with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ is infinite but less than the sets which have an empty intersection |∩{E(1), E(2), ...}| = 0, like the set of all endsegments. They are actually infinite. That is much more than all finite sets.

More delusional crap from the master.

Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:43:34 UTC+1:

> Every n ∈ ℕ is finite, and the set E(n) = {n, n+1, n+2, ...} has cardinality ℵo. So the set E is isomorphic to ℕ and hence has cardinality ℵo also.

What is the first natnumber residing in an infinite endsegment but not being an index?

Regards, WM

On Wednesday, 8 December 2021 at 07:39:43 UTC-4, WM wrote:
> Then name the last FISON. The first is {1}. If there is no last, then the sequence of FISONs is potentially infinite. That is so be definition. Shouting will not change this fact.

And the *SET* of all FISONs is actually infinite. Muckenheim, you are a pathetic liar.

> > the term
>
> was dropped in order to deceive newbies and others who are too stupid to recognize the fraud.
> > who insisted that physical reality must play a role in
> pure mathematics.
> Potential infinity has nothing to do wit reality.

Did you really say this??? Are you now so deranged that you can't even keep your attributions straight any more?

On Wednesday, 8 December 2021 at 07:49:11 UTC-4, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:43:34 UTC+1:
>
> > Every n ∈ ℕ is finite, and the set E(n) = {n, n+1, n+2, ...} has cardinality ℵo. So the set E is isomorphic to ℕ and hence has cardinality ℵo also.
> What is the first natnumber residing in an infinite endsegment but not being an index?

You truly are delusional. Why would you even think that there is such a thing? Or that it has anything to do with what you are so desperately trying to discuss here, with absolutely zero talent or success? Answer: You can't keep your quantifiers straight. Nothing to see here.

Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:57:14 UTC+1:
> On Wednesday, 8 December 2021 at 07:49:11 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:43:34 UTC+1:
> >
> > > Every n ∈ ℕ is finite, and the set E(n) = {n, n+1, n+2, ...} has cardinality ℵo. So the set E is isomorphic to ℕ and hence has cardinality ℵo also.
> > What is the first natnumber residing in an infinite endsegment but not being an index?
> Why would you even think that there is such a thing?

If all endsegments are infinite, then there is no exhaustion.

Regards, WM

Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:52:01 UTC+1:
> On Wednesday, 8 December 2021 at 07:39:43 UTC-4, WM wrote:
> > Then name the last FISON. The first is {1}. If there is no last, then the sequence of FISONs is potentially infinite. That is so be definition. Shouting will not change this fact.
> And the *SET* of all FISONs is actually infinite.

No.

But better this can be seen by the endsegments which have an infinite intersection.

Regards, WM

WM used his keyboard to write :
> FromTheRafters schrieb am Mittwoch, 8. Dezember 2021 um 11:39:19 UTC+1:
>> WM brought next idea :
>>>> (1) By ℵo-set you presumably mean a set of cardinality ℵo.
>>>
>>> In any case we can exclude that two consecutive ℵo-sets cannot exist in the
>>> natural order of ℕ.
>
>> Endsegments are taken
>> as infinite sets and your sequence of FISONs as sets are simply
>> infinitely many finite individual sets. "THERE IS NO POTENTIAL
>> INFINITY"
>
> Then name the last FISON.

There is no last FISON.

Learn some math, and stop lying.

WM formulated the question :
> Gus Gassmann schrieb am Mittwoch, 8. Dezember 2021 um 12:52:01 UTC+1:
>> On Wednesday, 8 December 2021 at 07:39:43 UTC-4, WM wrote:
>>> Then name the last FISON. The first is {1}. If there is no last, then the
>>> sequence of FISONs is potentially infinite. That is so be definition.
>>> Shouting will not change this fact.
>> And the *SET* of all FISONs is actually infinite.
>
> No.
>
> But better this can be seen by the endsegments which have an infinite
> intersection.

The only element which could possibly be in *every* endsegment set
would be the one which is the "last natural number" -- and since there
is no "last natural number" you should expect the emptyset to be the
intersection.

Unless you are stupid that is. Continuing to state this means you
either are that stupid or you are lying. Stop lying, I know the other
choice can't be helped.