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Nov 6, 2021, 5:42:36 PM11/6/21

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The following statements of about the set |N_F are all true

(i) Every element of |N_F is finite

(ii) |N_F, like any Peano set, has cardinality aleph_0

(iii) Between every element of |N_F and omega there are infinitely many natural numbers.

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

(iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

--

William Hughes

*i

(i) Every element of |N_F is finite

(ii) |N_F, like any Peano set, has cardinality aleph_0

(iii) Between every element of |N_F and omega there are infinitely many natural numbers.

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

(iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

--

William Hughes

*i

Nov 7, 2021, 6:56:59 AM11/7/21

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William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> The following statements of about the set |N_F are all true

>

> (i) Every element of |N_F is finite

>

> (ii) |N_F, like any Peano set, has cardinality aleph_0

>

> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

>

> (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?
> The following statements of about the set |N_F are all true

>

> (i) Every element of |N_F is finite

>

> (ii) |N_F, like any Peano set, has cardinality aleph_0

>

> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

>

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

>

> (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

>

Please explain with that model what elements are in all infinitely many infinite endsegments, the intersection of which is empty.
> (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

>

Regards, WM

Nov 7, 2021, 8:04:27 AM11/7/21

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On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > The following statements of about the set |N_F are all true

> >

> > (i) Every element of |N_F is finite

> >

> > (ii) |N_F, like any Peano set, has cardinality aleph_0

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

> > (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?

This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines. If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".
> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > The following statements of about the set |N_F are all true

> >

> > (i) Every element of |N_F is finite

> >

> > (ii) |N_F, like any Peano set, has cardinality aleph_0

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

> > (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?

> > (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

> >

> Please explain with that model what elements are in all infinitely many infinite endsegments, the intersection of which is empty.

Nov 7, 2021, 8:08:04 AM11/7/21

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On Sunday, November 7, 2021 at 7:56:59 AM UTC-4, WM wrote:

> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > The following statements of about the set |N_F are all true

> >

> > (i) Every element of |N_F is finite

> >

> > (ii) |N_F, like any Peano set, has cardinality aleph_0

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

> > (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. (N_F has no largest element).
> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > The following statements of about the set |N_F are all true

> >

> > (i) Every element of |N_F is finite

> >

> > (ii) |N_F, like any Peano set, has cardinality aleph_0

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

> > (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?

--

William Hughes

Nov 7, 2021, 2:24:20 PM11/7/21

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Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > The following statements of about the set |N_F are all true

> > >

> > > (i) Every element of |N_F is finite

> > >

> > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > >

> > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > >

> > > (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?

> This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

The ordinal line like the real line shows increasing values from left to right.
> On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > The following statements of about the set |N_F are all true

> > >

> > > (i) Every element of |N_F is finite

> > >

> > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > >

> > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > >

> > > (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?

> This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

> > > (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

> > >

> > Please explain with that model what elements are in all infinitely many infinite endsegments, the intersection of which is empty.

> This is easy: Every end segment E(n) contains infinitely many natural numbers, but never n - 1.

|∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

What gives

|∩{E(k) : k ∈ ℕ}| = 0 ?

Regards, WM

Nov 7, 2021, 2:27:31 PM11/7/21

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> (N_F has no largest element).

Regards, WM

Nov 7, 2021, 2:33:31 PM11/7/21

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On 11/7/2021 1:24 PM, WM wrote:

> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

>> On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

>>> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

>>>> The following statements of about the set |N_F are all true

>>>>

>>>> (i) Every element of |N_F is finite

>>>>

>>>> (ii) |N_F, like any Peano set, has cardinality aleph_0

>>>>

>>>> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

>>>>

>>>> (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?

>> This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

>

> The ordinal line like the real line shows increasing values from left to right.

>

>> If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

>

> Limits destroy set theory. If you cannot count all natural numbers then countability is nonsense. Therefore you have to count until only omega remains.

Your convoluted thinking has got you in a very small box. "Therefore you have to count until only omega remains."
> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

>> On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

>>> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

>>>> The following statements of about the set |N_F are all true

>>>>

>>>> (i) Every element of |N_F is finite

>>>>

>>>> (ii) |N_F, like any Peano set, has cardinality aleph_0

>>>>

>>>> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

>>>>

>>>> (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?

>> This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

>

> The ordinal line like the real line shows increasing values from left to right.

>

>> If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

>

> Limits destroy set theory. If you cannot count all natural numbers then countability is nonsense. Therefore you have to count until only omega remains.

>

>>>> (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

>>>>

>>> Please explain with that model what elements are in all infinitely many infinite endsegments, the intersection of which is empty.

>> This is easy: Every end segment E(n) contains infinitely many natural numbers, but never n - 1.

>

> If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

your equation has 2 errors, please correct them.

>

> What gives

> |∩{E(k) : k ∈ ℕ}| = 0 ?

>

> Regards, WM

>

Nov 7, 2021, 4:21:25 PM11/7/21

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On Sunday, 7 November 2021 at 15:24:20 UTC-4, WM wrote:

> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > The following statements of about the set |N_F are all true

> > > >

> > > > (i) Every element of |N_F is finite

> > > >

> > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > >

> > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > >

> > > > (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?

> > This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> The ordinal line like the real line shows increasing values from left to right.

So what, sirra, is the left-hand side of the ordinal line? I'd say, there is no better description than 1 (or 0 if you prefer).
> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > The following statements of about the set |N_F are all true

> > > >

> > > > (i) Every element of |N_F is finite

> > > >

> > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > >

> > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > >

> > > > (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?

> > This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> The ordinal line like the real line shows increasing values from left to right.

> > If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

> Limits destroy set theory. If you cannot count all natural numbers then countability is nonsense. Therefore you have to count until only omega remains.

> > > > (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.

> > > >

> > > Please explain with that model what elements are in all infinitely many infinite endsegments, the intersection of which is empty.

> > This is easy: Every end segment E(n) contains infinitely many natural numbers, but never n - 1.

> If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

>> 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, inclusion monotony and all other smokescreens you have thrown up over the years notwithstanding.
> What gives

> |∩{E(k) : k ∈ ℕ}| = 0 ?

Nov 7, 2021, 4:45:28 PM11/7/21

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Gus Gassmann schrieb am Sonntag, 7. November 2021 um 22:21:25 UTC+1:

> On Sunday, 7 November 2021 at 15:24:20 UTC-4, WM wrote:

> > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > > The following statements of about the set |N_F are all true

> > > > >

> > > > > (i) Every element of |N_F is finite

> > > > >

> > > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > > >

> > > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > > >

> > > > > (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?

> > > This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> > The ordinal line like the real line shows increasing values from left to right.

> So what, sirra, is the left-hand side of the ordinal line? I'd say, there is no better description than 1 (or 0 if you prefer).

The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?
> On Sunday, 7 November 2021 at 15:24:20 UTC-4, WM wrote:

> > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > > The following statements of about the set |N_F are all true

> > > > >

> > > > > (i) Every element of |N_F is finite

> > > > >

> > > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > > >

> > > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > > >

> > > > > (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?

> > > This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> > The ordinal line like the real line shows increasing values from left to right.

> So what, sirra, is the left-hand side of the ordinal line? I'd say, there is no better description than 1 (or 0 if you prefer).

> > > If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

> > Limits destroy set theory. If you cannot count all natural numbers then countability is nonsense. Therefore you have to count until only omega remains.

> As usual you try to shoehorn a natural language interpretation onto the term "countable".

Cantor made very clear that a. this does not work and b. what needs to be used instead.

> I am not going to trot all of this out again here.

> > If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

> You do not. You require a proof that ℕ_def is finite.

> You can't even supply that. (In fact, ℕ_def = ℕ and thus has cardinality ℵ₀.)

> >> 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, inclusion monotony and all other smokescreens

> > What gives

> > |∩{E(k) : k ∈ ℕ}| = 0 ?

> This is a trivial consequence of the fact that ℕ has cardinality ℵ₀.

Regards, WM

Nov 7, 2021, 4:51:35 PM11/7/21

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Regards, WM

Nov 7, 2021, 5:17:06 PM11/7/21

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William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

>

>

> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

>

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

How can that be? Every set of definable natnumbers reaches exactly as far as its elements: 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.
>

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

Regards, WM

Nov 7, 2021, 5:26:49 PM11/7/21

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On Sunday, 7 November 2021 at 17:45:28 UTC-4, WM wrote:

> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 22:21:25 UTC+1:

> > On Sunday, 7 November 2021 at 15:24:20 UTC-4, WM wrote:

> > > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > > > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

[...]
> Gus Gassmann schrieb am Sonntag, 7. November 2021 um 22:21:25 UTC+1:

> > On Sunday, 7 November 2021 at 15:24:20 UTC-4, WM wrote:

> > > Gus Gassmann schrieb am Sonntag, 7. November 2021 um 14:04:27 UTC+1:

> > > > On Sunday, 7 November 2021 at 07:56:59 UTC-4, WM wrote:

> > > > > What is left-hand side next to omega on the ordinal line?

> > > > This is totally unclear. "Left-hand side" refers to driving rules, or perhaps to equations, but not to ordinal lines.

> > > The ordinal line like the real line shows increasing values from left to right.

> > So what, sirra, is the left-hand side of the ordinal line? I'd say, there is no better description than 1 (or 0 if you prefer).

> The question is different from what you pretended to have understood: What is left-hand side next to omega on the ordinal line?

> > > > If you are asking what is the immediate predecessor of omega, the answer is: There is none. Omega does not have a predecessor, it is a "limit ordinal".

> > > Limits destroy set theory. If you cannot count all natural numbers then countability is nonsense. Therefore you have to count until only omega remains.

> > As usual you try to shoehorn a natural language interpretation onto the term "countable".

> No. ∀n ∈ ℕ is a clear and formal statement.

> Cantor made very clear that a. this does not work and b. what needs to be used instead.

> Yes, he used natural language because he had no formal yet: "eine solche gegenseitig eindeutige und vollständige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollständige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p. 415] In case of enumerating this means all natnural numbers are required until omega.

> > I am not going to trot all of this out again here.

> You know that you are wrong.

I truly know no such thing. What I *DO* know is that you have not presented *ONE* coherent argument in this forum. Trying to simply deny this without actually presenting a coherent argument strikes me as self-defeating.
> Yes, he used natural language because he had no formal yet: "eine solche gegenseitig eindeutige und vollständige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollständige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p. 415] In case of enumerating this means all natnural numbers are required until omega.

> > I am not going to trot all of this out again here.

> You know that you are wrong.

> > > If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

> > > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

> > You do not. You require a proof that ℕ_def is finite.

> No, I use what there is. ℕ_def is not finite but |ℕ_def| < ℵ₀ .

> > You can't even supply that. (In fact, ℕ_def = ℕ and thus has cardinality ℵ₀.)

> Look, if you collect only those endsegments which give an infinite intersection, then the cardinality cannot be ℵ₀.

> > >> 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, inclusion monotony and all other smokescreens

> > Inclusion monotony is so straight and simple that it cannot be denunciated as wrong by fools of matheology.

> > > What gives

> > > |∩{E(k) : k ∈ ℕ}| = 0 ?

> > This is a trivial consequence of the fact that ℕ has cardinality ℵ₀.

> 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. 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.
> > > |∩{E(k) : k ∈ ℕ}| = 0 ?

> > This is a trivial consequence of the fact that ℕ has cardinality ℵ₀.

> 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?

Nov 7, 2021, 5:29:33 PM11/7/21

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

> [...] 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", I'd say a simple reason is that omega is a limit ordinal.

Nov 7, 2021, 6:59:37 PM11/7/21

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WM presented the following explanation :

Proper subsets. Can you prove that they are natural numbers as you

claim they are? Can you show how the predecessor function connects each

element to the initial element, zero or one, your choice?

claim they are? Can you show how the predecessor function connects each

element to the initial element, zero or one, your choice?

Nov 8, 2021, 12:46:09 AM11/8/21

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Nov 8, 2021, 8:44:38 AM11/8/21

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On Sunday, November 7, 2021 at 3:27:31 PM UTC-4, WM wrote:

> William schrieb am Sonntag, 7. November 2021 um 14:08:04 UTC+1:

> > On Sunday, November 7, 2021 at 7:56:59 AM UTC-4, WM wrote:

> > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > The following statements of about the set |N_F are all true

> > > >

> > > > (i) Every element of |N_F is finite

> > > >

> > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > >

> > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > >

> > > > (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. |N_F, like any Peano set,
> William schrieb am Sonntag, 7. November 2021 um 14:08:04 UTC+1:

> > On Sunday, November 7, 2021 at 7:56:59 AM UTC-4, WM wrote:

> > > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > > > The following statements of about the set |N_F are all true

> > > >

> > > > (i) Every element of |N_F is finite

> > > >

> > > > (ii) |N_F, like any Peano set, has cardinality aleph_0

> > > >

> > > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > > >

> > > > (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.

does not have a last element.

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.

--

Wiliam Hughes

Nov 8, 2021, 8:50:02 AM11/8/21

to

On Sunday, November 7, 2021 at 6:17:06 PM UTC-4, WM wrote:

> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

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

> How can that be?

The set |N_F does not have a last element.
> William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> >

> > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> >

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

> How can that be?

--

William Hughes

Nov 8, 2021, 10:41:21 PM11/8/21

to

On 11/7/2021 3:51 PM, WM wrote:

> Serg io schrieb am Sonntag, 7. November 2021 um 20:33:31 UTC+1:

>> On 11/7/2021 1:24 PM, WM wrote:

>

>>> If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

>> "collect" is not a math term. Define what your "collect" means in terms of Math.

>

> If you don't know it, try to learn it without bothering me.

So you admit you cannot define it in terms of math nor equations.
> Serg io schrieb am Sonntag, 7. November 2021 um 20:33:31 UTC+1:

>> On 11/7/2021 1:24 PM, WM wrote:

>

>>> If you collect only endsegments which do not have an empty intersection together (collect as long as the intersection is infinite, then stop) then you get

>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

>> "collect" is not a math term. Define what your "collect" means in terms of Math.

>

> If you don't know it, try to learn it without bothering me.

>But it is not so important because 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?

>

> Regards, WM

>

Nov 9, 2021, 9:51:32 AM11/9/21

to

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 .

--

William Hughes

>[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,

>that is, they all contain infinitely many natural numbers each. What numbers are that?

--

William Hughes

Nov 9, 2021, 11:28:29 AM11/9/21

to

∀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

Nov 9, 2021, 11:35:47 AM11/9/21

to

William schrieb am Montag, 8. November 2021 um 14:50:02 UTC+1:

> On Sunday, November 7, 2021 at 6:17:06 PM UTC-4, WM wrote:

> > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > >

> > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > >

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

> > How can that be?

It is not. There are many natural numbers between "all definable natnumbers" ans omega.
> On Sunday, November 7, 2021 at 6:17:06 PM UTC-4, WM wrote:

> > William schrieb am Samstag, 6. November 2021 um 22:42:36 UTC+1:

> > >

> > > (iii) Between every element of |N_F and omega there are infinitely many natural numbers.

> > >

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

> > How can that be?

> The set |N_F does not have a last element.

Regards, WM

Nov 9, 2021, 11:47:19 AM11/9/21

to

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.
> 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.

> |N_F, like any Peano set,

> does not have a last element.

>

> 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. Between it and omega there are almost all aleph_0 natural numbers - thise which make all endsegments infinite although all Peano numbers are lost in the sequence of endsegments
> 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.

Regards, WM

Nov 9, 2021, 11:52:48 AM11/9/21

to

On Tuesday, November 9, 2021 at 12:47:19 PM UTC-4, WM wrote:

> 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.

Nope, whether or not you can write it down, It would still have to be the largest element of |N_F.
> 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.

>... set is potentially infinite.

Nonsense.

--

William Hughes

Nov 9, 2021, 11:55:26 AM11/9/21

to

FromTheRafters schrieb am Montag, 8. November 2021 um 00:59:37 UTC+1:

> WM presented the following explanation :

> 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.
> > 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.

> Can you prove that they are natural numbers as you

> claim they are?

> Can you show how the predecessor function connects each

> element to the initial element, zero or one, your choice?

∀k ∈ ℕ_def: E(k+1) = E(k) \ {k}

Nevertheless infinitely many natural numbers remain in *every* endsegment.

Regards, WM

Nov 9, 2021, 11:59:48 AM11/9/21

to

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. The fact that you cannot write a natural number down does not mean in is not in the order. "dark natural numbers" are not natural numbers. A natural number is an element of the Peano set |N_F.

--

William Hughes

>... natnumbers are dark and are not in a discernible order.

Nope, something that is not in the order is not a natural number. The fact that you cannot write a natural number down does not mean in is not in the order. "dark natural numbers" are not natural numbers. A natural number is an element of the Peano set |N_F.

--

William Hughes

Nov 9, 2021, 12:00:01 PM11/9/21

to

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 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:

--

William Hughes

Nov 9, 2021, 12:02:51 PM11/9/21

to

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.

>... infinitely many natural numbers remain in *every* endsegment.

But not the sane natural numbers.

Nov 9, 2021, 12:08:31 PM11/9/21

to

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.
> 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",

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

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

Nov 9, 2021, 12:19:36 PM11/9/21

to

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.
> 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 ("/\/\/\").

> > > 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".

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

> > 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.

> > > >> 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

> > 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.

∀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.

Regards, WM

Nov 9, 2021, 12:35:59 PM11/9/21

to

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.
> On Tuesday, November 9, 2021 at 12:55:26 PM UTC-4, WM wrote:

> >... infinitely many natural numbers remain in *every* endsegment.

>

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

Nov 9, 2021, 12:38:48 PM11/9/21

to

Regards, WM

Nov 9, 2021, 12:45:41 PM11/9/21

to

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.
> 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.

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

> "dark natural numbers" are not natural numbers.

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

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

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

Regards, WM

Nov 9, 2021, 12:49:15 PM11/9/21

to

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

> ...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

Nov 9, 2021, 12:56:25 PM11/9/21

to

Regards, WM

Nov 9, 2021, 12:58:51 PM11/9/21

to

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".

> 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

Nov 9, 2021, 1:04:14 PM11/9/21

to

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.
> 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

--

William Hughes

Nov 9, 2021, 1:09:36 PM11/9/21

to

On Tuesday, November 9, 2021 at 1:56:25 PM UTC-4, WM wrote:

> all

Ah the ambiguous "all".

--

William Hughes

> all

Ah the ambiguous "all".

--

William Hughes

Nov 9, 2021, 1:23:54 PM11/9/21

to

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.
> 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).

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

∀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.

Regards, WM

Nov 9, 2021, 1:51:47 PM11/9/21

to

On Tuesday, November 9, 2021 at 2:23:54 PM UTC-4, WM wrote:

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

> Why does the set differ so much from its elements?

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

Nov 9, 2021, 2:32:00 PM11/9/21

to

∀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

Nov 9, 2021, 2:41:23 PM11/9/21

to

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 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.

Nov 9, 2021, 2:44:25 PM11/9/21

to

WM brought next idea :

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.

'infinite' part of 'infinite endsegments'. It only shows that you like

to muddy the waters.

Nov 9, 2021, 2:46:07 PM11/9/21

to

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
> 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.

--

William Hughes

Nov 9, 2021, 2:51:57 PM11/9/21

to

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 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.

Nov 9, 2021, 2:55:16 PM11/9/21

to

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.
> 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?

[...]

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

[...]

> > > > 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".
> 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).

> > > 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).

Nov 10, 2021, 1:02:42 AM11/10/21

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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,
> 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:

> >

> > 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

Nov 10, 2021, 11:42:42 AM11/10/21

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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.
> 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

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.

> > 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".

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.

Regards, WM
Nov 10, 2021, 11:49:08 AM11/10/21

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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.
> 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.

> > 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.

> > 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

Nov 10, 2021, 12:46:04 PM11/10/21

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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?
> 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.

Nov 10, 2021, 4:00:53 PM11/10/21

to

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.
> 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.

(Your set |N is not a Peano set, the set of natural numbers is a Peano set).

--

William Hughes

Nov 11, 2021, 1:09:19 AM11/11/21

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Nov 11, 2021, 6:28:59 AM11/11/21

to

Set theorists confess that every natural number n has an infinite distance to ω

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

but that the set ℕ has no distance to ω. So either the distance is removed by some magic action when forming the set, or undefinable numbers are remaining and are simply included as individually unspecified elements in "the infinite set ℕ". These are the alternatives known to me. Are there more?

Regards, WM

Nov 11, 2021, 6:29:56 AM11/11/21

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zelos...@gmail.com schrieb am Donnerstag, 11. November 2021 um 07:09:19 UTC+1:

> onsdag 10 november 2021 kl. 17:42:42 UTC+1 skrev WM:

> > All visible numbers are restricted to ℕ_def in

> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.

>

> onsdag 10 november 2021 kl. 17:42:42 UTC+1 skrev WM:

> > All visible numbers are restricted to ℕ_def in

> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.

>

> >Impossible as long as |ℕ \ ℕ_def| = ℵ₀.

> Is false, given all definitions you have given for N_def, N_def=N

> Is false, given all definitions you have given for N_def, N_def=N

Nov 11, 2021, 6:32:04 AM11/11/21

to

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

The Peano set ℕ has no distance to ω. So either the distance is removed by some magic action when forming the set, or undefinable numbers are remaining and are simply included as individually unspecified elements in "the infinite Peano set ℕ". These are the alternatives known to me. Are there more?

Regards, WM

Nov 11, 2021, 7:35:05 AM11/11/21

to

On Thursday, 11 November 2021 at 07:28:59 UTC-4, WM wrote:

> Gus Gassmann schrieb am Mittwoch, 10. November 2021 um 18:46:04 UTC+1:

> > On Wednesday, 10 November 2021 at 12:49:08 UTC-4, WM wrote:

>

> > > 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?

> They represent some definable and all undefinable finite ordinals.

All natural numbers have been defined, hence are "definable". Your objection is risible.
> Gus Gassmann schrieb am Mittwoch, 10. November 2021 um 18:46:04 UTC+1:

> > On Wednesday, 10 November 2021 at 12:49:08 UTC-4, WM wrote:

>

> > > 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?

> They represent some definable and all undefinable finite ordinals.

> Set theorists confess that every natural number n has an infinite distance to ω

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

> but that the set ℕ has no distance to ω. So either the distance is removed by some magic action when forming the set, or undefinable numbers are remaining and are simply included as individually unspecified elements in "the infinite set ℕ". These are the alternatives known to me. Are there more?

Nov 11, 2021, 10:19:09 AM11/11/21

to

On Thursday, November 11, 2021 at 7:32:04 AM UTC-4, WM wrote:

> William schrieb am Mittwoch, 10. November 2021 um 22:00:53 UTC+1:

> > 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).

> But all [elements of a Peano set] n have an infinite distance to ω
> William schrieb am Mittwoch, 10. November 2021 um 22:00:53 UTC+1:

> > 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).

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

Nope, you have used your set |N which is not a Peano set. |N_F us a Peano set. What you want is
∀n ∈ |N_F: ||N_F\ {1, 2, 3, ..., n}| = ℵo

Indeed,. Note that ∀n ∈ |N_F: {1, 2, 3, ..., n} has a last element, this is a statement that is true for every element of the set |N_F. The set |N_F does not have a lase element.

> The Peano set ℕ has no distance to ω. So either the distance is removed by some magic action when forming the set, or undefinable numbers are remaining and are simply included as individually unspecified elements in "the infinite Peano set ℕ"

--

William Hughes

Nov 11, 2021, 1:34:44 PM11/11/21

to

On 11/9/2021 11:08 AM, WM wrote:

> 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.

wrong Cantor is dead, long time.

> But it seems that he has recognized that there is no gap: omega follows next upon the natural numbers.

your conjecture.

> 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.

wrong Cantor is dead, long time.

> But it seems that he has recognized that there is no gap: omega follows next upon the natural numbers.

your conjecture.

Nov 12, 2021, 4:01:50 AM11/12/21

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Nov 12, 2021, 4:37:26 PM11/12/21

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Gus Gassmann schrieb am Donnerstag, 11. November 2021 um 13:35:05 UTC+1:

> All natural numbers have been defined, hence are "definable".

Then this is a contradiction because of ∀k ∈ ℕ: E(k+1) = E(k) \ {k} .
> All natural numbers have been defined, hence are "definable".

∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo

~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

Regards, WM

Nov 12, 2021, 4:42:23 PM11/12/21

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William schrieb am Donnerstag, 11. November 2021 um 16:19:09 UTC+1:

> Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

Then your Peano set causes a contradiction:
> Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo

~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

Note that ∀k ∈ ℕ: E(k+1) = E(k) \ {k} .
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

Regards, WM

Nov 12, 2021, 5:34:32 PM11/12/21

to

On Friday, November 12, 2021 at 4:42:23 PM UTC-5, Transfinity wrote:

> William schrieb am Donnerstag, 11. November 2021 um 16:19:09 UTC+1:

>

> > Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

> Then your Peano set causes a contradiction:

You say something irrelevant about your set |N which is not a Peano set and is not the set of natural numbers. What you want is
> William schrieb am Donnerstag, 11. November 2021 um 16:19:09 UTC+1:

>

> > Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

> Then your Peano set causes a contradiction:

∀ n∈|N_F ∃ M(n)⊂|N_F: M(n)⊂E(n) ∧ |M(n)| = ℵo

Note that the set M(n) may change when the element of |N_F changes.

~∃ Q⊂|N_F ∀ n∈|N_F : Q⊂E(n) ∧ |Q| = ℵo .

No contradiction. For every element, n, of the set |N_F, there exits the set M(n) that depends on the element of |N_F.

There is no set, Q, that does not depend on the element of |N_F.

--

William Hughes

Nov 13, 2021, 4:22:51 AM11/13/21

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William schrieb am Freitag, 12. November 2021 um 23:34:32 UTC+1:

> On Friday, November 12, 2021 at 4:42:23 PM UTC-5, Transfinity wrote:

> > William schrieb am Donnerstag, 11. November 2021 um 16:19:09 UTC+1:

> >

> > > Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

> > Then your Peano set causes a contradiction:

> You say something irrelevant about your set |N which is not a Peano set and is not the set of natural numbers. What you want is

>

> ∀ n∈|N_F ∃ M(n)⊂|N_F: M(n)⊂E(n) ∧ |M(n)| = ℵo

>

> Note that the set M(n) may change when the element of |N_F changes.

It may but need not change. But importantly every set M(n) is infinite |M(n)| = ℵo.
> On Friday, November 12, 2021 at 4:42:23 PM UTC-5, Transfinity wrote:

> > William schrieb am Donnerstag, 11. November 2021 um 16:19:09 UTC+1:

> >

> > > Your ℕ is not a Peano set. A Peano set does not have "unspecified" (your latest way of saying dark) elements.

> > Then your Peano set causes a contradiction:

> You say something irrelevant about your set |N which is not a Peano set and is not the set of natural numbers. What you want is

>

> ∀ n∈|N_F ∃ M(n)⊂|N_F: M(n)⊂E(n) ∧ |M(n)| = ℵo

>

> Note that the set M(n) may change when the element of |N_F changes.

>

> ~∃ Q⊂|N_F ∀ n∈|N_F : Q⊂E(n) ∧ |Q| = ℵo .

>

> No contradiction. For every element, n, of the set |N_F, there exists the set M(n) that depends on the element of |N_F.
> ~∃ Q⊂|N_F ∀ n∈|N_F : Q⊂E(n) ∧ |Q| = ℵo .

>

That is said in the first statement already, not in the second. The second statement says that no infinite set M (you changed its name to Q but that does not change the contents of the statement) existst outside of |N_F - in fact not even one single element.

> There is no set, Q, that does not depend on the element of |N_F.

Regards, WM

Nov 13, 2021, 7:50:24 AM11/13/21

to

On Saturday, November 13, 2021 at 4:22:51 AM UTC-5, WM wrote:

> No infinite set

that does not change exists according to the second statement. There is an infinite set that changes

(i.e. a different infinite set for each element of |N_F) according to the first statement. No contradiction.

--

William Hughes

> No infinite set

that does not change exists according to the second statement. There is an infinite set that changes

(i.e. a different infinite set for each element of |N_F) according to the first statement. No contradiction.

--

William Hughes

Nov 13, 2021, 8:25:55 AM11/13/21

to

>∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo

>~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

Nov 13, 2021, 8:26:22 AM11/13/21