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

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.

Regards, WM

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.

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 Hughes

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 .

What gives
|∩{E(k) : k ∈ ℕ}| = 0 ?

Regards, WM

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. Then there is this point somewhere on the ordinal line. Then there is the question for next sensible.

> (N_F has no largest element).

N_F is followed by omega. Every definable element is not followed by omega.

Regards, WM

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."

>
>>>> (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 .

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

your equation has 2 errors, please correct them.

>
> What gives
> |∩{E(k) : k ∈ ℕ}| = 0 ?

what number is in every endsegments ? None.

>
> Regards, WM
>

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.

> > > > (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 .

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.) In addition, you studiously avoid talking about the nub of the problem:
>> 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 ?

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

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?

> > > 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.

> > 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?

Regards, WM

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

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.

Regards, WM

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:
[...]

> > > > > 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?

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 ("/\/\/\").

> > > > 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.

That is decidedly not what you wrote before. You nonsensically opined that "[i]f you cannot count all natural numbers then countability is nonsense".
> 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.

> > > 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 .

This is clearly about as daft as can be. There is no concept of "slightly infinite" or " almost infinite but not quite". (At least in ZFC there is not, but if you try to show ZFC inconsistent, you will have to prove two contradictory statements *WITHIN* ZFC. Since nothing you have done has come even close to showing even the most rudimentary understanding of the issues, I am not holding my breath that there will be anything substantive from you in the next, oh, omega years.)

> > 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.

Granted, but that is not using the set ℕ_def. How hard can that be to comprehend?

> > >> 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.

The level of indentation is again wrong. This line is your crap. I can assure you that I would never use the word "denunciated". Cute portmanteau, but nonetheless ridiculous.
> > > 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.

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.

WM presented the following explanation :
> 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?

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?

lördag 6 november 2021 kl. 22:42:36 UTC+1 skrev William:
> 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
so just N

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,
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

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 Hughes

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.

>But it is not so important because all endsegments have an empty intersection, that is, they do not contain any natural number in common,

you finally accept the proofs, and truth, good.

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

cant you answer your own question ?

>
> Regards, WM
>

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

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

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.

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

Neither does the set ℕ, but its elements touch omega. That is not a contradiction because most natnumbers are dark and are not in a discernible order. The simplest picture is the set of geometrical points of unit fractions. They are all there on the real line but there is no last before zero discernible.

Regards, WM

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. 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

Regards, WM

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.

>... set is potentially infinite.

Nonsense.

--
William Hughes

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.

> Can you show how the predecessor function connects each
> element to the initial element, zero or one, your choice?

It does not connect the dark natural numbers. All natural numbers connected by the successor function (which is tantamount with "being the last number of a FISON") get lost endsegment by endsegment:
∀k ∈ ℕ_def: E(k+1) = E(k) \ {k}
Nevertheless infinitely many natural numbers remain in *every* endsegment.

Regards, WM

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