onsdag 10 november 2021 kl. 17:42:42 UTC+1 skrev WM:
> William schrieb am Dienstag, 9. November 2021 um 20:46:07 UTC+1:
> > On Tuesday, November 9, 2021 at 3:32:00 PM UTC-4, WM wrote:
> > Every element of the set N_F heads a set that "much smaller" than a set with cardinality aleph_0. The set |N_F, like any Peano set, has cardinality aleph_0
> Impossible as long as |ℕ \ ℕ_def| = ℵ₀.
> Two consecutive infinite sets of ℵ₀ elements are impossible here.
> The instantaneous vanishing of |ℕ \ ℕ_def| is not mathematics but credo in absurdum.
> Therefore your claim is silly.
> > > Please observe logic:
> > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> > > That proves that *all* elements of ℕ_def of ℕ_F do not change the infinity of the remaining set of aleph_0 dark numbers.
> > |N_F does not contain "dark numbers".
> No, but ℕ does. All visible numbers are restricted to ℕ_def in
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> Regards, WM

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

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

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?

Regards, WM

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

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

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

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 Peano natural numbers n have an infinite distance to ω
∀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

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.

> 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?

Given your utter ignorance of mathematics, the answer is, of course there are more. You don't even for a moment consider that the "magic action" you invoke might be the simple taking of a limit, do you? How stupid does one have to be before Hochschule Augsburg considers one suitable for a professorship?

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
> ∀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"

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

--
William Hughes

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.

torsdag 11 november 2021 kl. 12:29:56 UTC+1 skrev WM:
> 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.
> >
> > >Impossible as long as |ℕ \ ℕ_def| = ℵ₀.
>
> > Is false, given all definitions you have given for N_def, N_def=N
> 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
Yes, the alternative which is the accurate is that you are a crank, an idiot, and you do not understand mathematics!

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} .
∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
Regards, WM

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 ∃ 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} .

Regards, WM

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 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

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.
>
> ~∃ 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.

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.

No infinite set outside of |N_F can be existing according to the second statement, but there is an infinite set outside of |N_F according to the first statement. This is a contradiction if |N_F is the same in both cases.

Regards, WM

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

fredag 12 november 2021 kl. 22:37:26 UTC+1 skrev WM:
> 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} .
>
> ∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
> ~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
>
> Regards, WM
>Then this is a contradiction because of ∀k ∈ ℕ: E(k+1) = E(k) \ {k} .
That causes no contradition.
>∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
NEITHER DOES THIS
>~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
You can just say ~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n)

fredag 12 november 2021 kl. 22:42:23 UTC+1 skrev Transfinity:
> 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 ∃ 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} .
>
> Regards, WM

Those are not contradictions

William schrieb am Samstag, 13. November 2021 um 13:50:24 UTC+1:
> On Saturday, November 13, 2021 at 4:22:51 AM UTC-5, WM wrote:
>

∀ n∈ℕ_def ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

> > No infinite set
> that does not change exists according to the second statement. There is an infinite set that changes

Irrelevant. It remains infinite.
> (i.e. a different infinite set for each element of |N_F) according to the first statement.

E(n) differs. M is the fixed subset ℵo. That means natural numbers can only be chosen from ℕ\M which has not cardinality ℵo, because there are not two sets of ℵo in the natural order of ℕ.

> No contradiction.

The second statement covers all naturakl numbers, more than the first statement.

Regards, WM

On Saturday, November 13, 2021 at 1:05:59 PM UTC-5, WM wrote:
> (i.e. a different infinite set for each element of |N_F) according to the first statement.
> E(n) differs. M is the fixed subset ℵo

There is no fixed subset.

> That means natural numbers can only be chosen from ℕ\M

Nope you are using your set |N which is not a Peano set and M which does not exist. This is nonsense.

--
William Hughes

William schrieb am Samstag, 13. November 2021 um 19:19:12 UTC+1:
> On Saturday, November 13, 2021 at 1:05:59 PM UTC-5, WM wrote:
> > (i.e. a different infinite set for each element of |N_F) according to the first statement.
> > E(n) differs. M is the fixed subset ℵo
> There is no fixed subset.

That is irrelevant, as long all sets are infinite. Why should it be of any relevance? Can you tell me? In every case natural numbers can only be chosen from ℕ\M which is a finite set. Or do you believe in two actually infinite concecutive sets in ℕ? Where does the second one start?

> Nope you are using your set |N which is not a Peano set and M which does not exist.

I am not using it but ℕ which exists like M according to ZF
∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo .

It turns out however that ℕ causes a contradiction: Between all n∈ℕ and ω there are infinitely many natnumbers. The second statement contradicts this one:
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
There is nothimg between ℕ and ω. A contradiction.

Note that the set of positive fractions smaller than (n+1)/n is infinite. It has a fixed an infinite minimum. Same with M.

Regards, WM

zelos...@gmail.com schrieb am Samstag, 13. November 2021 um 14:26:22 UTC+1:
> > ∀ n∈ℕ ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
> > ~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
> Those are not contradictions
A simply unfounded and wrong claim.
Between all n∈ℕ and ω there are infinitely many natnumbers according to the first line. There is nothing between ℕ and ω according to
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| > 0.
A contradiction.
Regards, WM

On Saturday, November 13, 2021 at 2:02:03 PM UTC-5, WM wrote:
> ℕ\M which is a finite set

|N is not the natural numbers and M does not exist. Nonsense.

--
William Hughes

William schrieb am Samstag, 13. November 2021 um 20:10:41 UTC+1:
> On Saturday, November 13, 2021 at 2:02:03 PM UTC-5, WM wrote:
> > ℕ\M which is a finite set
> |N is not the natural numbers and M does not exist.

ℕ ist the genuine set of natural numbers defined by Zermelo or v. Neumann or Peano.

Again: Why do you deny that every M, even if not constant but never smaller than ℵo, when subtracted from this genuine ℕ restricts the set ℕ\M to be finite?

Note that every set of positive fractions smaller than (n+1)/n is infinite. It has a fixed an infinite minimum. Same with M. But M contains ℵo natural numbers. Hence |ℕ\M| < ℵo for every possible M.

Regards, WM

WM laid this down on his screen :
> William schrieb am Samstag, 13. November 2021 um 20:10:41 UTC+1:
>> On Saturday, November 13, 2021 at 2:02:03 PM UTC-5, WM wrote:
>>> ℕ\M which is a finite set
>>> N is not the natural numbers and M does not exist.
>
> ℕ ist the genuine set of natural numbers defined by Zermelo or v. Neumann or
> Peano.

Finally, you admit that the natural numbers are defined.

On Saturday, November 13, 2021 at 2:21:25 PM UTC-5, WM wrote:
> William schrieb am Samstag, 13. November 2021 um 20:10:41 UTC+1:
> > On Saturday, November 13, 2021 at 2:02:03 PM UTC-5, WM wrote:
> > > ℕ\M which is a finite set
> > |N is not the natural numbers and M does not exist.
> ℕ ist the genuine set of natural numbers

Nope. the set of natural numbers is a Peano set. Your |N is not a Peano set.
Any element of M has to be an element of E(n) for every n in |N_F. There are no such elements

--
William Hughes

crank Wolfgang Mückenheim, aka WM wrote:
> William schrieb am Samstag, 13. November 2021 um 13:50:24 UTC+1:
>> On Saturday, November 13, 2021 at 4:22:51 AM UTC-5, WM wrote:
>>
>
> ∀ n∈ℕ_def ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
> ~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .
>
>>> No infinite set
>> that does not change exists according to the second statement. There is an infinite set that changes
>
> Irrelevant. It remains infinite.
>
>> (i.e. a different infinite set for each element of |N_F) according to the first statement.
>
> E(n) differs. M is the fixed subset ℵo.

ℵo is not a subset of N, crank Wolfgang Mückenheim, from Hochschule
Augsburg. In ∀ n∈ℕ_def ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo, M cannot be fixed
(i.e. independant of n)

FromTheRafters schrieb am Samstag, 13. November 2021 um 20:30:28 UTC+1:
> WM laid this down on his screen :
> > William schrieb am Samstag, 13. November 2021 um 20:10:41 UTC+1:
> >> On Saturday, November 13, 2021 at 2:02:03 PM UTC-5, WM wrote:
> >>> ℕ\M which is a finite set
> >>> N is not the natural numbers and M does not exist.
> >
> > ℕ ist the genuine set of natural numbers defined by Zermelo or v. Neumann or
> > Peano.
> Finally, you admit that the natural numbers are defined.

The natural numbers are defined collectively. But most are not defined individually. The difference is clearly visible here:

∀ n∈ℕ_def ∃ M⊂ℕ: M⊂E(n) ∧ |M| = ℵo
~∃ M⊂ℕ ∀ n∈ℕ : M⊂E(n) ∧ |M| = ℵo .

Regards, WM