Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
> On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
>
> > All infinitely many FISONs (1, 2, 3, ..., n)
> Learn to use standard notation,
>
> (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e IN].

The first is an ordered set, the second pretends to be an ordered set, although the curly brackets destroy any order, such that in principle the set could differ severely from the intended meaning.

Regards, WM

WM submitted this idea :
> Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
>> On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
>>
>>> All infinitely many FISONs (1, 2, 3, ..., n)
>> Learn to use standard notation,
>>
>> (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e
>> IN].
>
> The first is an ordered set,

It looks more like a finite sequence.

On Thursday, December 23, 2021 at 6:28:35 AM UTC-4, WM wrote:
> And I dare to talk about potentially infinite sets

A potentially infinite object changes. A set does not change. Potentially infinite set is nonsense. Yes or no?

--
William Hughes

On Thursday, December 23, 2021 at 11:32:40 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
> > On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
> >
> > > All infinitely many FISONs (1, 2, 3, ..., n)
> > Learn to use standard notation,
> >
> > (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e IN].
> >
> The first

Shut up, idiot!

"{1, 2, 3, ..., n}" denotes a FISON, namely the set {m e IN : m <= n}, but "(1, 2, 3, ..., n)" doesn't.

Holy shit!!!

On 12/23/2021 4:28 AM, WM wrote:
> William schrieb am Mittwoch, 22. Dezember 2021 um 20:00:10 UTC+1:
>> On Wednesday, December 22, 2021 at 2:01:30 PM UTC-4, WM wrote:
>
>>>>>> Each element, n, of the set of natural numbers if finite. There is no largest n. The set of natural numbers is infinite.
>>>>> Potentially infinite
>>>> The natural numbers are a Peano set. "Potentially infinite set" is nonsense.
>> You keep saying "Potentially infinite" when talking about a set. "Potentially infinite set" is nonsense. Yes or no?
>
> From the standpoint of set theory sets must be actually infinite.

wrong, sets can be infinite. not "must be".

>From the standpoint of German DIN-norm and even of the EU, zero is a natural number.

wrong, that is dated, and it depends upon the context. Zero is a whole number.

> I dare to tell my students that I do not accept zero as a natural number because it is one of the most unnatural numbers and I recommend that they should join me.

Sad you do not explain both sides and demand they be subservient to you.

> And I dare to talk about potentially infinite sets because actually infinite sets are mostly dark.

Dark to you, because you do not have the skills in math to see the light.

> Sue me.

We have started a "Go Fund Me Page" to sue you and your WM's Troll Math.
We now have HUGE SUMS of money donated by your ex students.
Given the potential gravity of the situation, that actual money got spent on a non-potential actual doughnut this morning to go with the coffee.

Your math is on trial in this news group, and since you refuse to support it with mathematical Proofs, and basing it on your imagination, we suggest you
hire potentially more lawyers for your actual potential trial.

>
> Regards, WM

FromTheRafters schrieb am Donnerstag, 23. Dezember 2021 um 14:53:43 UTC+1:
> WM submitted this idea :
> > Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
> >> On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
> >>
> >>> All infinitely many FISONs (1, 2, 3, ..., n)
> >> Learn to use standard notation,
> >>
> >> (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e
> >> IN].
> >
> > The first is an ordered set,
> It looks more like a finite sequence.

It is both.

Cantor says:

(a1, a2, ... a, ...),

(a1, a2, ... a, ... b1, b2, ..., b, ...),

(a1, a2, ... a, ... b1, b2, ..., b, ... c1, c2, c3)

are well-ordered sets. He uses parentheses. Why should I use different symbols? Because some of his disciples have made an unauthorized decision?

Regards, WM

William schrieb am Donnerstag, 23. Dezember 2021 um 15:15:51 UTC+1:
> On Thursday, December 23, 2021 at 6:28:35 AM UTC-4, WM wrote:
> > And I dare to talk about potentially infinite sets
> A potentially infinite object changes. A set does not change. Potentially infinite set is nonsense. Yes or no?

No. The nonsense follows in the dark portion.

All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there

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

What happens when subtracting the union of FISONs? There are two alternatives:
(1) Either also almost all natnumbers remain. This would prove that they are dark. So dark numbers exist.
(2) Or no natnumbers remain. Then these natnumbers can be subtracted collectively (by the union of FISONs) but not individually (by any FISON). These natnumbers are dark.
Therefore both alternatives prove the existence of dark numbers.

Regards, WM

Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 15:23:00 UTC+1:
> On Thursday, December 23, 2021 at 11:32:40 AM UTC+1, WM wrote:

> "{1, 2, 3, ..., n}" denotes a FISON, namely the set {m e IN : m <= n}, but "(1, 2, 3, ..., n)" doesn't.
>
Nehmen wir als Beispiel die wohlgeordnete Menge

F = (a1, a2, ... a, ... b1, b2, ..., b, ... c1, c2, c3),

so sind hier durch das Element a3 der Abschnitt

(a1, a2)

und der zugehörige Rest

(a3, a4, ... a+2, ... b1, b2, ..., b, ... c1, c2, c3),

Georg Cantor, the only authority, the inventor.

Regards, WM

On 12/23/2021 10:43 AM, WM wrote:
> FromTheRafters schrieb am Donnerstag, 23. Dezember 2021 um 14:53:43 UTC+1:
>> WM submitted this idea :
>>> Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
>>>> On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
>>>>
>>>>> All infinitely many FISONs (1, 2, 3, ..., n)
>>>> Learn to use standard notation,
>>>>
>>>> (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e
>>>> IN].
>>>
>>> The first is an ordered set,
>> It looks more like a finite sequence.
>
> It is both.

Wrong.

>
> Cantor says:

wrong, Cantor said (he is dead)

>
> (a1, a2, ... a, ...),
>
> (a1, a2, ... a, ... b1, b2, ..., b, ...),
>
> (a1, a2, ... a, ... b1, b2, ..., b, ... c1, c2, c3)

got any references or are you just making stuff up again?

>
> are well-ordered sets.

URL please.

> He uses parentheses.

perhaps because printers did not have {} back then

> Why should I use different symbols?

Why do you continue to make mistakes ?

> Because some of his disciples have made an unauthorized decision?

so you reject all of modern Math, as it is "an unauthorized decisions by (Cantors) disciples".

your words, not mine.

>
> Regards, WM

on 12/23/2021, WM supposed :
> FromTheRafters schrieb am Donnerstag, 23. Dezember 2021 um 14:53:43 UTC+1:
>> WM submitted this idea :
>>> Fritz Feldhase schrieb am Donnerstag, 23. Dezember 2021 um 03:48:05 UTC+1:
>>>> On Wednesday, December 22, 2021 at 7:01:30 PM UTC+1, WM wrote:
>>>>
>>>>> All infinitely many FISONs (1, 2, 3, ..., n)
>>>> Learn to use standard notation,
>>>>
>>>> (1, 2, 3, ..., n) is *not* a FISON, but {1, 2, 3, ..., n} is [for any n e
>>>> IN].
>>>
>>> The first is an ordered set,
>> It looks more like a finite sequence.
>
> It is both.
>
> Cantor says:
>
> (a1, a2, ... a, ...),
>
> (a1, a2, ... a, ... b1, b2, ..., b, ...),
>
> (a1, a2, ... a, ... b1, b2, ..., b, ... c1, c2, c3)
>
> are well-ordered sets. He uses parentheses. Why should I use different
> symbols?

To stay current. Oh, yeah, I see your point, why should you.

On 12/23/2021 10:47 AM, WM wrote:
> William schrieb am Donnerstag, 23. Dezember 2021 um 15:15:51 UTC+1:
>> On Thursday, December 23, 2021 at 6:28:35 AM UTC-4, WM wrote:
>>> And I dare to talk about potentially infinite sets
>> A potentially infinite object changes. A set does not change. Potentially infinite set is nonsense. Yes or no?
>
> No.

wrong.

> The nonsense follows in the dark portion.

correct

>
> All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there

wrong, corrected:

The natural numbers, ℕ, can be divided into 2 sets, A FISON(n) (1, 2, 3, ..., n) and an endsegment(n+1).

>
<snip>
>
> What happens when subtracting the union of FISONs?

subtract union of which FISONs ?

>There are two alternatives:
> (1) Either also almost all natnumbers remain.

Which FISONs? why "almost"? why "remain"?

> This would prove that they are dark. So dark numbers exist.

no, it does not prove anything.

> (2) Or no natnumbers remain.

wrong. fail. fail to specify which FISONS are in the union.

> Then these natnumbers can be subtracted collectively (by the union of FISONs) but not individually (by any FISON). These natnumbers are dark.

fail. you are in the dark.

> Therefore both alternatives prove the existence of dark numbers.

you have only your incorrect statements, no Proof. Fail.

>
> Regards, WM

On Thursday, December 23, 2021 at 12:47:19 PM UTC-4, WM wrote:
> William schrieb am Donnerstag, 23. Dezember 2021 um 15:15:51 UTC+1:
> > On Thursday, December 23, 2021 at 6:28:35 AM UTC-4, WM wrote:
> > > And I dare to talk about potentially infinite sets
> > A potentially infinite object changes. A set does not change. Potentially infinite set is nonsense. Yes or no?
> No.

WM syllogism

A set does not change.
A potentially infinite object does change
A set can be a potentially infinite object

>The nonsense follows in the dark portion.

A Peano set has cardinaliy aleph_0 and no dark elements (trivial induction)

--
William Hughes

William schrieb am Donnerstag, 23. Dezember 2021 um 22:59:52 UTC+1:

> A Peano set has cardinaliy aleph_0 and no dark elements (trivial induction)

The set of FISONs is a Peano set.

All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there
∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
What happens when subtracting the union of FISONs? There are two alternatives:
(1) Either also almost all natnumbers remain. This would prove that they are dark. So dark numbers exist.
(2) Or no natnumbers remain. Then these natnumbers can be subtracted collectively (by the union of FISONs) but not individually (by any FISON). These natnumbers are dark.
Therefore both alternatives prove the existence of dark numbers.

ℕ is not a Peano set. (trivial proof)

Regards, WM

On Friday, December 24, 2021 at 7:09:25 AM UTC-4, WM wrote:
> William schrieb am Donnerstag, 23. Dezember 2021 um 22:59:52 UTC+1:
>
> > A Peano set has cardinaliy aleph_0 and no dark elements (trivial induction)
> The set of FISONs is a Peano set.

Indeed, the set of FISONs has cardinatlity aleph_0 and no dark elements (trivial induction).

> All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there

Nope, "all infintely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.

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

This is a statement about the elements of the set of FISONs. As you note above, "infinitely many", there are and infinite number of elements in the set of FISONs.

--
William Hughes

William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
> On Friday, December 24, 2021 at 7:09:25 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 23. Dezember 2021 um 22:59:52 UTC+1:
> >
> > > A Peano set has cardinaliy aleph_0 and no dark elements (trivial induction)
> > The set of FISONs is a Peano set.
> Indeed, the set of FISONs has cardinatlity aleph_0 and no dark elements (trivial induction).
> > All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there
> Nope, "all infintely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.

Then the set or union differs from all its elements, i.e., from all definable natural numbers.
>
> > ∀n ∈ ℕ_P: |ℕ_P \ {1, 2, 3, ..., n}| = ℵo .
>
> This is a statement about the elements of the set of FISONs.

About *all* its elements. i.e., about all definable numbers.

> As you note above, "infinitely many", there are and infinite number of elements in the set of FISONs

which, according to your opinion, differ from this set by an infinite number of dark natural numbers.

Regards, WM

On Friday, December 24, 2021 at 11:22:08 AM UTC-4, WM wrote:
> William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:

> > Nope, "all infinitely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.
> Then the set or union differs from all its elements

Nope "all its elements" is the set. The set differs from each of its elements.
Each element of the set, a Peano set, is finite. The set, a Peano set, is infinite.

--
William Hughes

On 12/24/2021 9:22 AM, WM wrote:
> William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
>> On Friday, December 24, 2021 at 7:09:25 AM UTC-4, WM wrote:
>>> William schrieb am Donnerstag, 23. Dezember 2021 um 22:59:52 UTC+1:
>>>
>>>> A Peano set has cardinaliy aleph_0 and no dark elements (trivial induction)
>>> The set of FISONs is a Peano set.
>> Indeed, the set of FISONs has cardinatlity aleph_0 and no dark elements (trivial induction).
>>> All infinitely many FISONs (1, 2, 3, ..., n), when subtracted from ℕ, leave almost all natural numbers there
>> Nope, "all infintely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.
>
> Then the set or union differs from all its elements, i.e., from all definable natural numbers.

sets, unions, elements are different, that is why we use different words for each.

"definable numbers" however is imagination

>>
>>> ∀n ∈ ℕ_P: |ℕ_P \ {1, 2, 3, ..., n}| = ℵo .
>>
>> This is a statement about the elements of the set of FISONs.
>
> About *all* its elements. i.e., about all definable numbers.

there is none of that in the equation above

>
>> As you note above, "infinitely many", there are and infinite number of elements in the set of FISONs
>
> which, according to your opinion, differ from this set by an infinite number of dark natural numbers.

So where are your darkies at ?

>
> Regards, WM
>

William schrieb am Freitag, 24. Dezember 2021 um 17:30:48 UTC+1:
> On Friday, December 24, 2021 at 11:22:08 AM UTC-4, WM wrote:
> > William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
> > > Nope, "all infinitely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.
> > Then the set or union differs from all its elements
> Nope "all its elements" is the set.

All FISONs are an infinite sequence and all FISONs, when subtracted from ℕ, leave ℵ₀ natnumbers. Whether you deny it or not is irrelevant. It is a simple fact.

> The set differs from each of its elements.
> Each element of the set, a Peano set, is finite. The set, a Peano set, is infinite.

Nevertheless the sequence of all FISONs, which can also be called the set of all FISONs, when subtracted from ℕ, leaves ℵ₀ natnumbers. They cannot be defined, addressed, used as individuals for counting or in mappings.

Regards, WM

After serious thinking WM wrote :
> William schrieb am Freitag, 24. Dezember 2021 um 17:30:48 UTC+1:
>> On Friday, December 24, 2021 at 11:22:08 AM UTC-4, WM wrote:
>>> William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
>>>> Nope, "all infinitely many FISONs", the set of FISONs, when subtracted
>>>> from N_P leaves nothing.
>>> Then the set or union differs from all its elements
>> Nope "all its elements" is the set.
>
> All FISONs are an infinite sequence

All FISONs *is* an infinite set.

On Saturday, 25 December 2021 at 07:09:37 UTC-4, FromTheRafters wrote:
> After serious thinking WM wrote :
> > All FISONs are an infinite sequence
> All FISONs *is* an infinite set.

Not really. "All FISONs" is ambiguous nonsense (no offense!).

On Saturday, December 25, 2021 at 6:21:04 AM UTC-4, WM wrote:
> William schrieb am Freitag, 24. Dezember 2021 um 17:30:48 UTC+1:
> > On Friday, December 24, 2021 at 11:22:08 AM UTC-4, WM wrote:
> > > William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
> > > > Nope, "all infinitely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.
> > > Then the set or union differs from all its elements
> > Nope "all its elements" is the set.
> All FISONs are an infinite sequence

The set of FISONs is ordered and infinite, so the set o FISONs is also an infinite sequence.
As every element of the set of natural numbers is the largest element of a FISON (trivial induction), when you subtract the union of all elements of the infinite -sequence of FISONs from the natural numbes you get the empty set.

>and all FISONs, when subtracted from ℕ, leave ℵ₀ natnumbers.

Nope you are confusing "all FISONs", the infinite sequence of FISONs, with the elements of the infinite sequence of FISONs.

--
William Hughes

On 12/25/2021 4:20 AM, WM wrote:
> William schrieb am Freitag, 24. Dezember 2021 um 17:30:48 UTC+1:
>> On Friday, December 24, 2021 at 11:22:08 AM UTC-4, WM wrote:
>>> William schrieb am Freitag, 24. Dezember 2021 um 15:02:15 UTC+1:
>>>> Nope, "all infinitely many FISONs", the set of FISONs, when subtracted from N_P leaves nothing.
>>> Then the set or union differs from all its elements
>> Nope "all its elements" is the set.
>
> All FISONs are an infinite sequence

no. The set of all FISONs is an infinite sequence.

you confuse elements and sets.

William schrieb am Samstag, 25. Dezember 2021 um 17:23:11 UTC+1:
> On Saturday, December 25, 2021 at 6:21:04 AM UTC-4, WM wrote:

> > All FISONs are an infinite sequence
> The set of FISONs is ordered and infinite, so the set o FISONs is also an infinite sequence.
> As every element of the set of natural numbers is the largest element of a FISON (trivial induction), when you subtract the union of all elements of the infinite -sequence of FISONs from the natural numbes you get the empty set.

Maybe. But every FISON, when individually subtracted from ℕ, leaves ℵ₀ natnumbers. Therefore the union or sequence or set of all FISONs, when subtracted from ℕ, leaves less - nothing as you say.. This proves: By collectively subtracting, more is subtracted than by individually subtracting.

> >and all FISONs, when subtracted from ℕ, leave ℵ₀ natnumbers.
> Nope you are confusing "all FISONs", the infinite sequence of FISONs, with the elements of the infinite sequence of FISONs.

Only the elements which can be treated individually will leave ℵo numbers:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
The infinite sequence as collection or set does not leave anything.

Now we have fixed the clear distinction between natnumbers which can be handled individually and the infinite dark remainder.

Regards, WM

On Sunday, December 26, 2021 at 11:05:31 AM UTC-4, WM wrote:
>... By collectively subtracting, more is subtracted than by individually subtracting.

Collectively subtracting,: subtracting an infinite set.
Individually subtracting,: subtracting a finite set.
Indeed, the two are different,
However the Peano set of natural numbers does not contain dark elements.

--
William Hughes

On 12/20/2021 11:28 AM, WM wrote:
> Chris M. Thomasson schrieb am Montag, 20. Dezember 2021 um 09:48:17 UTC+1:
>> On 12/19/2021 1:52 AM, WM wrote:
>>
>> It goes on forever. There is no end.
>
> There is nothing beyond omega. And all natnumbers before omega can be exhausted in bijections if set theory is correct.

You seem to want to confuse infinite with finite, on purpose? Humm... Troll?