On Tuesday, November 9, 2021 at 12:28:29 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 15:51:32 UTC+1:
> > On Sunday, November 7, 2021 at 5:51:35 PM UTC-4, WM wrote:
> > >[the set of]l endsegments [has] an empty intersection, that is, they do not contain any natural number in common,
> >
> > correct.
> > >but nevertheless all endsegments are infinite,
> > Correct. This follows from an element of the set |N_F is finite, the set |N_F is infinite.
> > >that is, they all contain infinitely many natural numbers each. What numbers are that?
> > A different set for each .
> Name some elements which are not removed by the intersection:
There are no elements of |N_F that are not "removed by the intersection". There is no element that in in every one of the different sets.

--
William Hughes

On Tuesday, November 9, 2021 at 12:55:26 PM UTC-4, WM wrote:
>... infinitely many natural numbers remain in *every* endsegment.

But not the sane natural numbers.

Gus Gassmann schrieb am Sonntag, 7. November 2021 um 23:29:33 UTC+1:
> On Sunday, 7 November 2021 at 18:17:06 UTC-4, WM wrote:
> > [...] Yes, between every definable element and omega there are infinitely many natural numbers. There is a reason however, why the properties of infinite sets differ from the properties of their elements and why sets reach farther, until omega. Guess why.
>
> Aside from the fact that sets do not "reach",

Sets of ordinal numbers have an extension on the ordinal line.

> I'd say a simple reason is that omega is a limit ordinal.

Cantor does not know that the set of natnumbers must reach until omega because he only knows definable numbers. But it seems that he has recognized that there is no gap: omega follows next upon the natural numbers.

Oder aber die Zahlen β enthalten keine größte, dann besitzen sie (nach dem zweiten Erzeugungsprinzip) eine "Grenze" β', welche auf alle β zunächst folgt, (Cantor, p. 208f).

Regards, WM

Gus Gassmann schrieb am Sonntag, 7. November 2021 um 23:26:49 UTC+1:
> On Sunday, 7 November 2021 at 17:45:28 UTC-4, WM wrote:

> > The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?
> As I said: You do not know how to express yourself properly. However, the answer is equally easy: three dots ("...") or perhaps a squiggly or zigzagging line ("/\/\/\").

What comes before the three dots? Note that we are in actual infinity.

> > > As usual you try to shoehorn a natural language interpretation onto the term "countable".
> > No. ∀n ∈ ℕ is a clear and formal statement.
> That is decidedly not what you wrote before. You nonsensically opined that "[i]f you cannot count all natural numbers then countability is nonsense"..

So it is.

> if you try to show ZFC inconsistent, you will have to prove two contradictory statements *WITHIN* ZFC.

It is sufficient to show that dark numbers are implied by actual infinity.

> > Look, if you collect only those endsegments which give an infinite intersection, then the cardinality cannot be ℵ₀.
> Granted, but that is not using the set ℕ_def.

There is no set but only a potentially infinite collection.

> > > >> Every end segment E(n) contains infinitely many natural numbers, but never n - 1.
> > > The intersection of all end segments *CAN* be (and (*IS*) empty, even though each end segment has cardinality ℵ₀

So it is . This proves that no numbers are staying in all endsegments and infinitely many numbers are staying in all endsegments (because there are none entering the scene). This is a contradiction if all are definable. But obviously they are not.

> > You should try to understand! All definable numbers are lost but all endsegments with empty intersection contain infinitely many natnumbers. Which numbers remain when the intersection is empty?
> Several things about this mishmash of ideas: First, nothing is ever "lost" from an intersection.

Step by step the sequece of intersections decreases:
∀k ∈ ℕ: E(k+1) = E(k) \ {k}
element by element.

> Second, the intersection contains exactly those numbers that are contained in every end segment. Since there are infinitely many end segments E(n), and no end segment E(n+1) contains n, the intersection is empty. In other words, it does not contain anything, and in particular it doesn't contain numbers.

What do all infinite endsegments contain? What natnumbers are in every endsegment but do not contribute to the intersection of all endsegments (which otherwise would be infinite).

Regards, WM

William schrieb am Dienstag, 9. November 2021 um 18:02:51 UTC+1:
> On Tuesday, November 9, 2021 at 12:55:26 PM UTC-4, WM wrote:
> >... infinitely many natural numbers remain in *every* endsegment.
>
> But not the same natural numbers.

Name only some of your choice in any endsegment of your choice which are not lost by the mechanism proving that all are lost from the intersection:
∀k ∈ ℕ: E(k+1) = E(k) \ {k} .
Since not all are lost from the endsegments, there must infinitely many others remain in every infinite endsegment.

Regards, WM

William schrieb am Dienstag, 9. November 2021 um 18:00:01 UTC+1:
> On Tuesday, November 9, 2021 at 12:28:29 PM UTC-4, WM wrote:
> > Name some elements which are not removed by the intersection:
> There are no elements of |N_F that are not "removed by the intersection". There is no element that in in every one of the different sets.

Yes. But infinitely many natural numbers remain in every endsegment, in all endsegments of the infinite sequence. I claim that these natural numbers are dark. Can you identify one of them?

Regards, WM

William schrieb am Dienstag, 9. November 2021 um 17:59:48 UTC+1:
> On Tuesday, November 9, 2021 at 12:35:47 PM UTC-4, WM wrote:
> >... natnumbers are dark and are not in a discernible order.
>
> Nope, something that is not in the order is not a natural number.

It is not a definable natural number, and it is certainly not a familiar object. But without them there is no actually infinite set ℕ.

> The fact that you cannot write a natural number down does not mean in is not in the order.

There is a complete set ℕ filling the ordinal line between 0 and ω or, if you insist, some point before omega.

> "dark natural numbers" are not natural numbers.

They are not definable natural numbers, mostly.

> A natural number is an element of the Peano set |N_F.

For all these numbers we get

∀n ∈ ℕ_Peano: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

It is strictly excluded by logic that they are actually infinite.

Regards, WM

On Tuesday, November 9, 2021 at 1:35:59 PM UTC-4, WM wrote:

> ...Since not all are lost from the endsegments,

Nope. One element of |N_F is "lost" from every endsegment (a different element of each endsegment). There are an infinite number of endsegments. All are "lost from the endsegments".

--
William Hughes

William schrieb am Dienstag, 9. November 2021 um 18:49:15 UTC+1:
> On Tuesday, November 9, 2021 at 1:35:59 PM UTC-4, WM wrote:
>
> > ...Since not all are lost from the endsegments,
>
> Nope. One element of |N_F is "lost" from every endsegment (a different element of each endsegment). There are an infinite number of endsegments. All are "lost from the endsegments".

But all endsegments remain infinite. What blows them up if all definable elements are lost?

Regards, WM

On Tuesday, November 9, 2021 at 1:38:48 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 18:00:01 UTC+1:
> > On Tuesday, November 9, 2021 at 12:28:29 PM UTC-4, WM wrote:
>
> > > Name some elements which are not removed by the intersection:
> > There are no elements of |N_F that are not "removed by the intersection".
> Yes. But infinitely many natural numbers remain in every endsegment

Indeed,
(iii) Between every element of |N_F and omega there is an infinite set of natural numbers (a different set for each element).

However, there is no element that in in every one of the different sets.

(iv) There is no natural number between the set |N_F and omega.

--
William Hughes

On Tuesday, November 9, 2021 at 1:45:41 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 17:59:48 UTC+1:
> > On Tuesday, November 9, 2021 at 12:35:47 PM UTC-4, WM wrote:
> > >... natnumbers are dark and are not in a discernible order.
> >
> > Nope, something that is not in the order is not a natural number.
> It is not a definable natural number,

Indeed, it is not any type of natural number.

> and it is certainly not a familiar object. But without them there is no actually infinite set ℕ.
> > The fact that you cannot write a natural number down does not mean in is not in the order.
> There is a complete set ℕ filling the ordinal line between 0 and ω or, if you insist, some point before omega.
> > "dark natural numbers" are not natural numbers.
> They are not definable natural numbers

Indeed, they are not any type of natural number.

--
William Hughes

On Tuesday, November 9, 2021 at 1:56:25 PM UTC-4, WM wrote:
> all
Ah the ambiguous "all".

--
William Hughes

William schrieb am Dienstag, 9. November 2021 um 18:58:51 UTC+1:
> On Tuesday, November 9, 2021 at 1:38:48 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 9. November 2021 um 18:00:01 UTC+1:
> > > On Tuesday, November 9, 2021 at 12:28:29 PM UTC-4, WM wrote:
> >
> > > > Name some elements which are not removed by the intersection:
> > > There are no elements of |N_F that are not "removed by the intersection".
> > Yes. But infinitely many natural numbers remain in every endsegment
> Indeed,
> (iii) Between every element of |N_F and omega there is an infinite set of natural numbers (a different set for each element).

It is this one: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ...., n}| = ℵo.

The sets are not very different however, because almost all elements, namely ℵo, remain the same.
>
> However, there is no element that in in every one of the different sets.

There are almost all elements the same in these sets, because finite is much, much smaller than infinite. How could an infinite set be reduced to zero/empty in finite steps? Steps which according to
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
never touch the infinite majority?
Your claim therefore is blatantly wrong.

> (iv) There is no natural number between the set |N_F and omega.

Why does the set differ so much from its elements? Your claim unwittingly establishes dark numbers. They are what is required to make the properties of the set differ from the properties of all finite subsets of its definable elements.

Regards, WM

On Tuesday, November 9, 2021 at 2:23:54 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 18:58:51 UTC+1:

> > (iv) There is no natural number between the set |N_F and omega.
> Why does the set differ so much from its elements?

It does not "differ so much", However no element of |N_F is infinite. The set |N_F is infinite, not because it contains
an infinite element, but because, like any Peano set, it has no last element (Note, |N_F does not contain any "dark" elements but it is infinte)

--
William Hughes

William schrieb am Dienstag, 9. November 2021 um 19:51:47 UTC+1:
> On Tuesday, November 9, 2021 at 2:23:54 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 9. November 2021 um 18:58:51 UTC+1:
>
> > > (iv) There is no natural number between the set |N_F and omega.
> > Why does the set differ so much from its elements?
> It does not "differ so much", However no element of |N_F is infinite. The set |N_F is infinite, not because it contains
> an infinite element, but because, like any Peano set, it has no last element (Note, |N_F does not contain any "dark" elements but it is infinte)

It is much smaller however than every set of aleph_0 elements. Please observe logic:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
That proves that *all* elements of ℕ_def of ℕ_F do not change the infinity of the remaining set of aleph_0 dark numbers. Your impression that ℕ_def could be actually infinite is simply wrong and contradicts the above statement, i.e., the simplest logic. If you consciously violate logic there is no reason to continue. If you did unconsciously, then it's time to recognize the truth.

Regards, WM

WM pretended :
> FromTheRafters schrieb am Montag, 8. November 2021 um 00:59:37 UTC+1:
>> WM presented the following explanation :
>>> all endsegments have an empty intersection, that is, they
>>> do not contain any natural number in common, but nevertheless all
>>> endsegments are infinite, that is, they all contain infinitely many
>>> natural numbers each. What numbers are that?
>> Proper subsets.
>
> Of course. But no element can be defined.
>
>> Can you prove that they are natural numbers as you
>> claim they are?
>
> Endsegments contain only natural numbers.

That is not a proof outside of muckymath.

WM brought next idea :
> William schrieb am Dienstag, 9. November 2021 um 18:02:51 UTC+1:
>> On Tuesday, November 9, 2021 at 12:55:26 PM UTC-4, WM wrote:
>>> ... infinitely many natural numbers remain in *every* endsegment.
>>
>> But not the same natural numbers.
>
> Name only some of your choice in any endsegment of your choice which are not
> lost by the mechanism proving that all are lost from the intersection: ∀k ∈
> ℕ: E(k+1) = E(k) \ {k} . Since not all are lost from the endsegments, there
> must infinitely many others remain in every infinite endsegment.

Until you have defined finite endsegments, you should leave out the
'infinite' part of 'infinite endsegments'. It only shows that you like
to muddy the waters.

On Tuesday, November 9, 2021 at 3:32:00 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 19:51:47 UTC+1:
> > On Tuesday, November 9, 2021 at 2:23:54 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 9. November 2021 um 18:58:51 UTC+1:
> >
> > > > (iv) There is no natural number between the set |N_F and omega.
> > > Why does the set differ so much from its elements?
> > It does not "differ so much", However no element of |N_F is infinite. The set |N_F is infinite, not because it contains
> > an infinite element, but because, like any Peano set, it has no last element (Note, |N_F does not contain any "dark" elements but it is infinte)
> It is much smaller however than every set of aleph_0 elements.

Piffle. Every element of the set N_F heads a set that "much smaller" than a set with cardinality aleph_0. The set |N_F, like any Peano set, has cardinality aleph_0

> Please observe logic:
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> That proves that *all* elements of ℕ_def of ℕ_F do not change the infinity of the remaining set of aleph_0 dark numbers.

|N_F does not contain "dark numbers".

--
William Hughes

on 11/9/2021, WM supposed :
> William schrieb am Montag, 8. November 2021 um 14:44:38 UTC+1:
>> On Sunday, November 7, 2021 at 3:27:31 PM UTC-4, WM wrote:
>
>>>>>> (iv) There is no natural number between the set |N_F and omega.
>>>>> What is left-hand side next to omega on the ordinal line?
>>>> "next to omega on the left-hand side" is meaningless.
>>> It is not meaningless if omega exists.
>> Nope, An ordinal "next to omega on the left hand side" would have to be the
>> largest element of |N_F.
>
> If it were definable.
>
>>> N_F, like any Peano set,
>> does not have a last element.
>
> But omega is not arbitrarily far from the natnumbers. It is following next
> upon all natnumbers, the "der Größe nach zunächst folgende Zahl" (Cantor). So
> there is no gap. And if it were, then we could ask where it starts.
>>
>> Recall our starting assumption, "|N_F is a Peano set". You need to show this
>> assumption leads to a contradiction. not to results (e.g. there is no
>> natural number "next to omega on the left hand side") you do not like.
>
> The Peano set is potentially infinite.

Stop lying.

On Tuesday, 9 November 2021 at 13:19:36 UTC-4, WM wrote:
> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 23:26:49 UTC+1:
> > On Sunday, 7 November 2021 at 17:45:28 UTC-4, WM wrote:
>
> > > The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?
> > As I said: You do not know how to express yourself properly. However, the answer is equally easy: three dots ("...") or perhaps a squiggly or zigzagging line ("/\/\/\").
> What comes before the three dots?

A finite number of integers. Any finite set will do.

[...]
> > if you try to show ZFC inconsistent, you will have to prove two contradictory statements *WITHIN* ZFC.
> It is sufficient to show that dark numbers are implied by actual infinity..

Yeah, well, good luck with that.

[...]
> > > > The intersection of all end segments *CAN* be (and (*IS*) empty, even though each end segment has cardinality ℵ₀
> So it is . This proves that no numbers are staying in all endsegments and infinitely many numbers are staying in all endsegments (because there are none entering the scene).

I will comment on this natural language word salad. It is so bad, it is "not even wrong".
> > > You should try to understand! All definable numbers are lost but all endsegments with empty intersection contain infinitely many natnumbers. Which numbers remain when the intersection is empty?
> > Several things about this mishmash of ideas: First, nothing is ever "lost" from an intersection.
> Step by step the sequece of intersections decreases:
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> element by element.

Forming the intersection is not a sequential process, no matter how hard that is for you to comprehend.

> > Second, the intersection contains exactly those numbers that are contained in every end segment. Since there are infinitely many end segments E(n), and no end segment E(n+1) contains n, the intersection is empty. In other words, it does not contain anything, and in particular it doesn't contain numbers.
> What do all infinite endsegments contain? What natnumbers are in every endsegment but do not contribute to the intersection of all endsegments (which otherwise would be infinite).

Oh *SHUT* *UP*, you blithering idiot. No natural number n is an element of the end segment E(n+1). We had this, and it is tiresome to have to correct your moronic mistakes every single time. Muckenheim: Not a single coherent thought since 2005.

tisdag 9 november 2021 kl. 17:28:29 UTC+1 skrev WM:
> William schrieb am Dienstag, 9. November 2021 um 15:51:32 UTC+1:
> > On Sunday, November 7, 2021 at 5:51:35 PM UTC-4, WM wrote:
> > >[the set of]l endsegments [has] an empty intersection, that is, they do not contain any natural number in common,
> >
> > correct.
> > >but nevertheless all endsegments are infinite,
> > Correct. This follows from an element of the set |N_F is finite, the set |N_F is infinite.
> > >that is, they all contain infinitely many natural numbers each. What numbers are that?
> > A different set for each .
> Name some elements which are not removed by the intersection:
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> You know: There remains only the empty set. But nevertheless all endsegments are infinite. There remains an infinite set. May it be different for each endsegment. Name only some natnumbers which are not removed.
>
> Regards, WM
why is this the hill you die on? This is so stupid that you go on and on about this

William schrieb am Dienstag, 9. November 2021 um 20:46:07 UTC+1:
> On Tuesday, November 9, 2021 at 3:32:00 PM UTC-4, WM wrote:
> Every element of the set N_F heads a set that "much smaller" than a set with cardinality aleph_0. The set |N_F, like any Peano set, has cardinality aleph_0

Impossible as long as |ℕ \ ℕ_def| = ℵ₀.
Two consecutive infinite sets of ℵ₀ elements are impossible here.
The instantaneous vanishing of |ℕ \ ℕ_def| is not mathematics but credo in absurdum.
Therefore your claim is silly.
> > Please observe logic:
> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> > That proves that *all* elements of ℕ_def of ℕ_F do not change the infinity of the remaining set of aleph_0 dark numbers.
> |N_F does not contain "dark numbers".

No, but ℕ does. All visible numbers are restricted to ℕ_def in
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.

Regards, WM

Gus Gassmann schrieb am Dienstag, 9. November 2021 um 20:55:16 UTC+1:
> On Tuesday, 9 November 2021 at 13:19:36 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 23:26:49 UTC+1:
> > > On Sunday, 7 November 2021 at 17:45:28 UTC-4, WM wrote:
> >
> > > > The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?
> > > As I said: You do not know how to express yourself properly. However, the answer is equally easy: three dots ("...") or perhaps a squiggly or zigzagging line ("/\/\/\").
> > What comes before the three dots?
> A finite number of integers. Any finite set will do.

No, the ordinal line does not change upon your order.

> > Step by step the sequece of intersections decreases:
> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > element by element.
> Forming the intersection is not a sequential process, no matter how hard that is for you to comprehend.

That implies the existence of dark parts of the sequence.

> > What do all infinite endsegments contain? What natnumbers are in every endsegment but do not contribute to the intersection of all endsegments (which otherwise would be infinite).
> No natural number n is an element of the end segment E(n+1).

That is well known. You need not repeat it over and over again. The question is: What is remaining to make all endsegments infinite? And what is passing them? (Every infinite endsegment, even every non-empty endsegment is passed by some natnumbers.)

Regards, WM

On Wednesday, 10 November 2021 at 12:49:08 UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 9. November 2021 um 20:55:16 UTC+1:
> > On Tuesday, 9 November 2021 at 13:19:36 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 23:26:49 UTC+1:
> > > > On Sunday, 7 November 2021 at 17:45:28 UTC-4, WM wrote:
> > >
> > > > > The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?
> > > > As I said: You do not know how to express yourself properly. However, the answer is equally easy: three dots ("...") or perhaps a squiggly or zigzagging line ("/\/\/\").
> > > What comes before the three dots?
> > A finite number of integers. Any finite set will do.
> No, the ordinal line does not change upon your order.

Oh please. There are infinitely many natural numbers, and there is no largest. Why the hell do you think you need the three dots in the first place?

On Wednesday, November 10, 2021 at 12:42:42 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 9. November 2021 um 20:46:07 UTC+1:
> > On Tuesday, November 9, 2021 at 3:32:00 PM UTC-4, WM wrote:
> > Every element of the set N_F heads a set that "much smaller" than a set with cardinality aleph_0. The set |N_F, like any Peano set, has cardinality aleph_0
> Impossible as long as |ℕ \ ℕ_def| = ℵ₀.

The finite set of natural numbers that can be written down, |N_def, is irrelevant as is your set |N which is not the set of natrural numbers.
(Your set |N is not a Peano set, the set of natural numbers is a Peano set)..

--
William Hughes