On Monday, December 6, 2021 at 5:43:11 AM UTC-5, WM wrote:
> potentially infinite.

Potentially infinite set is nonsense.

> never addressed natnumbers

You have a large number of ways of saying "cannot be written down"

--
William Hughes

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.
>
> Regards, WM
no, it is INFINITE.

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

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

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>

Look, you silly crank. In set theory sets are either finite or infinite (i. e. not finite).

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

On Tuesday, December 7, 2021 at 12:57:36 PM UTC+1, WM wrote:
> William schrieb am Dienstag, 7. Dezember 2021 um 02:05:31 UTC+1:
> >
> > "Potentially infinite set" is nonsense.
> >
> Every <bla>

"Potentially infinite set" is nonsense.

Hint: In set theory sets are either finite or infinite (i. e. not finite).

On Tuesday, 7 December 2021 at 07:57:36 UTC-4, WM wrote:
[...]

> "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?

No interpretation. Your statement is insane. A finite set of end segments is just that, a finite set of end segments, namely a *SUBSET* of the set of *ALL* end segments, S, which is isomorphic to ℕ. So the "collection" of "[e]very finite set of endsegments" is a subset of P(S), the power set of S.

A "potentially infinite collection" is the same thing as a potentially infinite set, that is, an oxy, moron.

On Tuesday, December 7, 2021 at 1:18:14 PM UTC+1, WM wrote:
> 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.

*lol*

You really are an idiot, Mückenheim, you know.

JVR schrieb am Dienstag, 7. Dezember 2021 um 13:26:03 UTC+1:
> On Tuesday, December 7, 2021 at 1:01:14 PM UTC+1, WM wrote:

> > 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 interpretation?

> (2) You will readily understand why you can't define this concept if you try to explain what you mean by the intersection of 2 or

Look above. There you have a potentiall infinite set or better collection. The actually infinite set has empty intersection. This implies that this set has more elements - ℵ₀ more elements to be precise.

Regards, WM

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

--
William Hughes

On 12/7/2021 5:57 AM, WM wrote:
> 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

that is false.

> then
> |∩{E(1), E(2), ..., E(n+1)}| = ℵ₀

that is false

> because
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀

false

> This shows that the collection of collected infinite endsegments with infinite intersection is infinite. Do you agree?

false

>
> "Every finite set of endsegments" is the name of a potentially infinite *false* collection of infinite endsegments. Do you agree? If not, what is your interpretatition?

it is WMs bad math

>
> Regards, WM
>

On 12/7/2021 6:01 AM, 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:
>>>> 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?

all of it is obviously wrong.

>
> Regards, WM
>

On 12/7/2021 6:18 AM, WM wrote:
> 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.

that is what Your math is, "completely useless nonsense with no relation to logic", well said.

>
> Regards, WM
>

On 12/7/2021 7:53 AM, WM wrote:
> JVR schrieb am Dienstag, 7. Dezember 2021 um 13:26:03 UTC+1:
>> On Tuesday, December 7, 2021 at 1:01:14 PM UTC+1, WM wrote:
>
>>> 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 interpretation?
>
>> (2) You will readily understand why you can't define this concept if you try to explain what you mean by the intersection of 2 or
>
> Look above. There you have a potentiall infinite set or better collection.

by potentially, you mean you stopped at n that is all. when you do, it is a finite set.

> The actually infinite set has empty intersection. This implies that this set has more elements - ℵ₀ more elements to be precise.

no, it is just bad math.

>
> Regards, WM
>

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.

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.

Regards, WM

Gus Gassmann schrieb am Dienstag, 7. Dezember 2021 um 13:28:32 UTC+1:
> On Tuesday, 7 December 2021 at 07:57:36 UTC-4, WM wrote:
> [...]
> > "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?
> No interpretation. Your statement is insane. A finite set of end segments is just that, a finite set of end segments, namely a *SUBSET* of the set of *ALL* end segments, S, which is isomorphic to ℕ. So the "collection" of "[e]very finite set of endsegments" is a subset of P(S), the power set of S.

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.

> A "potentially infinite collection" is the same thing as a potentially infinite set, that is, an oxy, moron.

Obviously it exists. All endsegments have an empty intersection. The potentially infinite collection of infinite endsegments has an infinite intersection.

Regards, WM

> (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.

Exercise: Prove that the set E of endsegments E(n) with |E(n)| = ℵo cannot have |E| = ℵo.
Hint: If in doubt how to begin, try to find the first natnumber of contents remaining in all endsegments.

Regards, WM

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".

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

> Regards, WM
>

On 12/7/2021 11:00 AM, WM wrote:
>
>> (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.

ℵo + ℵo = ℵo

On 12/7/2021 11:05 AM, Fritz Feldhase wrote:
> 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".
>

(E|n|) < |E(n)| for all k in B, where B is undefined

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.

On Tuesday, 7 December 2021 at 12:52:35 UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 7. Dezember 2021 um 13:28:32 UTC+1:
> > On Tuesday, 7 December 2021 at 07:57:36 UTC-4, WM wrote:
> > [...]
> > > "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?
> > No interpretation. Your statement is insane. A finite set of end segments is just that, a finite set of end segments, namely a *SUBSET* of the set of *ALL* end segments, S, which is isomorphic to ℕ. So the "collection" of "[e]very finite set of endsegments" is a subset of P(S), the power set of S.
> 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.

Enough already with your mind-numbing category errors. 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.

> > A "potentially infinite collection" is the same thing as a potentially infinite set, that is, an oxy, moron.
> 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. Your hallucinations are getting weirder and more troublesome by the day. What they don't do is buy you a clue.