On Wednesday, January 11, 2023 at 7:41:42 PM UTC+1, Sergi o wrote:
> On 1/11/2023 12:11 PM, Fritz Feldhase wrote:

> > *** For WM ***
> >
> > IN = {1, 2, 3, ...}
> >
> > F_1 = {1}
> > F_2 = {1, 2}
> > etc.
> >
> > E_1 = {1, 2, 3, ...}
> > E_2 = {2, 3, 4, ...}
> > etc.
> >
> > Not that elegant. On the other hand, not wrong either.
> >
> so, E_1 ∩ E_2 = F_1 = {1}

Is it? Rather E_1 ∩ E_2 = E_2 = {2, 3, 4, ...}

> and E_2 ∩ E_3 = F_2 ∪ F_1 = {2}

Is it?Rather E_2 ∩ E_3 = E_3 = {3, 4, 5, ...} and F_1 ∪ F_2 = F_2 = {1, 2}

More errors:

> and E_3 ∩ E_4 = F_3 ∪ F_4 = {3}
>
> or
> E_(k) ∩ E_(k+1) = F_(k) ∪ F_(k+1) = {k}
>

Sergi o brought next idea :
> On 1/11/2023 12:11 PM, Fritz Feldhase wrote:
>> On Wednesday, January 11, 2023 at 5:35:00 PM UTC+1, FromTheRafters wrote:
>>> WM wrote on 1/10/2023 :
>>>> FromTheRafters schrieb am Dienstag, 10. Januar 2023 um 20:53:14 UTC+1:
>>>>>
>>>>> Where is the FISON associated with the initial element?
>>
>> *** For WM ***
>>
>> IN = {1, 2, 3, ...}
>>
>> and
>>
>> F_1 = {1}
>> F_2 = {1, 2}
>> etc.
>>
>>>> {1}
>>>>
>>> How can this be if E(1) = IN ?
>>
>> E_1 = {1, 2, 3, ...}
>> E_2 = {2, 3, 4, ...}
>> etc.
>>
>> And right. Hence NOT E_n = IN \ F_1 (in this context).
>>
>> Just deal with it. :-P
>>
>> Not that elegant. On the other hand, not wrong either.
>
>
> so, E_1 ∩ E_2 = F_1 = {1}
>
> and E_2 ∩ E_3 = F_2 ∪ F_1 = {2}
>
> and E_3 ∩ E_4 = F_3 ∪ F_4 = {3}
>
> or
> E_(k) ∩ E_(k+1) = F_(k) ∪ F_(k+1) = {k}
>
>
> now we're countin !
>
> (for completness F_0 = {}, so above E_1 ∩ E_2 = F_1 ∪ F_0 , so far no need
> for E_0)

If a FISON end is n then the associated endsegment beginning is n+1
If a FISON end is n-1 then the associated endsegment beginning is n

There is always a difference of one between the FISON end and the
endsegment index.

F(1) implies E(2) which is not his set of naturals. E(1)'s initial
element, has no predecessors and as such no FISON or rather the
emptyset or zero -- but there is no zero.

On Wednesday, January 11, 2023 at 8:03:00 PM UTC+1, FromTheRafters wrote:

> >> *** For WM ***
> >>
> >> IN = {1, 2, 3, ...}
> >>
> >> and
> >>
> >> F_1 = {1}
> >> F_2 = {1, 2}
> >> etc.
> >>
> >> E_1 = {1, 2, 3, ...}
> >> E_2 = {2, 3, 4, ...}
> >> etc.
> >>
> If a FISON end is n then the associated endsegment beginning is n+1

Yes. Hence: An e IN: F_n u E_(n+1) = IN.

> If a FISON end is n-1 then the associated endsegment beginning is n

Yes. Hence: An e IN \ {1}: F_(n-1) u E_n = IN.

> There is always a difference of one between the FISON end and the
> endsegment index.

Indeed.

> F(1) implies E(2) which is not his set of naturals. E(1)'s initial
> element, has no predecessors and as such no FISON or rather the
> emptyset or zero -- but there is no zero.

Yes. It's a mess, isn't it?

How about IN = {0, 1, 2, 3, ...} and

F_n := {m e IN: m < n} (for all n e IN)

E_n := {m e IN: m >= n} (for all n e IN)

?

Then:

F(0) = { } and E(0) = IN and F(0) n E(0) = { } and F(0) u E(0) = IN.

And for all n e IN \ {0}:

F(n) = {0, 1, ..., n-1} and E(n) = {n, n+1, n+2, ...} and F(n) n E(n) = {} and F(n) u E(n) = IN.

On 1/11/2023 12:48 PM, Fritz Feldhase wrote:
> On Wednesday, January 11, 2023 at 7:41:42 PM UTC+1, Sergi o wrote:
>> On 1/11/2023 12:11 PM, Fritz Feldhase wrote:
>
>>> *** For WM ***
>>>
>>> IN = {1, 2, 3, ...}
>>>
>>> F_1 = {1}
>>> F_2 = {1, 2}
>>> etc.
>>>
>>> E_1 = {1, 2, 3, ...}
>>> E_2 = {2, 3, 4, ...}
>>> etc.
>>>
>>> Not that elegant. On the other hand, not wrong either.
>>>
>> so, E_1 ∩ E_2 = F_1 = {1}
>
> Is it? Rather E_1 ∩ E_2 = E_2 = {2, 3, 4, ...}
>
>> and E_2 ∩ E_3 = F_2 ∪ F_1 = {2}
>
> Is it?Rather E_2 ∩ E_3 = E_3 = {3, 4, 5, ...} and F_1 ∪ F_2 = F_2 = {1, 2}
>
> More errors:
>
>> and E_3 ∩ E_4 = F_3 ∪ F_4 = {3}
>>
>> or
>> E_(k) ∩ E_(k+1) = F_(k) ∪ F_(k+1) = {k}
>>

agree, wrong operator......

On Wednesday, January 11, 2023 at 8:24:35 PM UTC+1, Fritz Feldhase wrote:

> How about IN = {0, 1, 2, 3, ...} and
>
> F_n := {m e IN: m < n} (for all n e IN)
>
> E_n := {m e IN: m >= n} (for all n e IN)
>
> ?
>
> Then:
>
> F(0) = { } and E(0) = IN and F(0) n E(0) = { } and F(0) u E(0) = IN.
>
> And for all n e IN \ {0}:
>
> F(n) = {0, 1, ..., n-1} and E(n) = {n, n+1, n+2, ...} and F(n) n E(n) = {} and F(n) u E(n) = IN.

If IN in our context is defined due to von Neumann, then IN coincides with the_set_of_FISONs := {F(n) : n e IN} = {{ }, {0}, {0, 1}, ...} = {0, 1, 2, ...}.

A simple and clean approach (in contrast to WM's "approach").

Why is { } a "finite initial segement of natural numbers"? Well, it's certainly a finite set. Moreover it's a segement of natural numbers: each and every element in { } is a natural numbers, and all natural numbers in { } are "connected" (there are no gaps between them).

On Wednesday, January 11, 2023 at 8:36:42 PM UTC+1, Fritz Feldhase wrote:
> On Wednesday, January 11, 2023 at 8:24:35 PM UTC+1, Fritz Feldhase wrote:
>
> > How about IN = {0, 1, 2, 3, ...} and
> >
> > F_n := {m e IN: m < n} (for all n e IN)
> >
> > E_n := {m e IN: m >= n} (for all n e IN)
> >
> > ?
> >
> > Then:
> >
> > F(0) = { } and E(0) = IN and F(0) n E(0) = { } and F(0) u E(0) = IN.
> >
> > And for all n e IN \ {0}:
> >
> > F(n) = {0, 1, ..., n-1} and E(n) = {n, n+1, n+2, ...} and F(n) n E(n) = {} and F(n) u E(n) = IN.
> If IN in our context is defined due to von Neumann, then IN coincides with the_set_of_FISONs := {F(n) : n e IN} = {{ }, {0}, {0, 1}, ...} = {0, 1, 2, ...}.
>
> A simple and clean approach (in contrast to WM's "approach").
>
> Why is { } a "finite initial segement of natural numbers"? Well, it's certainly a finite set. Moreover it's a segement of natural numbers: each and every element in { } is a natural numbers, and all natural numbers in { } are "connected" (there are no gaps between them).

And it's an _initial_ segement of natural numbers since for each and every element n in { } all natural numbers smaller than n are in { } too.

On Wednesday, January 11, 2023 at 8:36:42 PM UTC+1, Fritz Feldhase wrote:
> >
> > How about IN = {0, 1, 2, 3, ...} and
> >
> > F_n := {m e IN: m < n} (for all n e IN)
> >
> > E_n := {m e IN: m >= n} (for all n e IN)
> >
> > ?
> >
> > Then:
> >
> > F(0) = { } and E(0) = IN and F(0) n E(0) = { } and F(0) u E(0) = IN.
> >
> > And for all n e IN \ {0}:
> >
> > F(n) = {0, 1, ..., n-1} and E(n) = {n, n+1, n+2, ...} and F(n) n E(n) = {} and F(n) u E(n) = IN.
> >
> If IN in our context is defined due to von Neumann, then IN coincides with the_set_of_FISONs := {F(n) : n e IN} = {{ }, {0}, {0, 1}, ...} = {0, 1, 2, ...}.
>
> A simple and clean approach (in contrast to WM's "approach").

WM: "I am denying that it is meaningful to talk about the set of all n or the set of all FISONs.
This set does not exist as a completed entity. [You can easily see this by the observation that
otherwise there must be an infinite finite segment of IN or an infinite sequence of 1's must have
the limit 2.]"

Source: https://www.hs-augsburg.de/homes/mueckenh/Transfinity/KB/KB%20801-1000.pdf

Yeah, we can easily see that this guy is a mathematica crank par excellence.

Fritz Feldhase explained :
> On Wednesday, January 11, 2023 at 8:24:35 PM UTC+1, Fritz Feldhase wrote:
>
>> How about IN = {0, 1, 2, 3, ...} and
>>
>> F_n := {m e IN: m < n} (for all n e IN)
>>
>> E_n := {m e IN: m >= n} (for all n e IN)
>>
>> ?
>>
>> Then:
>>
>> F(0) = { } and E(0) = IN and F(0) n E(0) = { } and F(0) u E(0) = IN.
>>
>> And for all n e IN \ {0}:
>>
>> F(n) = {0, 1, ..., n-1} and E(n) = {n, n+1, n+2, ...} and F(n) n E(n) = {}
>> and F(n) u E(n) = IN.
>
> If IN in our context is defined due to von Neumann, then IN coincides with
> the_set_of_FISONs := {F(n) : n e IN} = {{ }, {0}, {0, 1}, ...} = {0, 1, 2,
> ...}.
>
> A simple and clean approach (in contrast to WM's "approach").
>
> Why is { } a "finite initial segement of natural numbers"? Well, it's
> certainly a finite set. Moreover it's a segement of natural numbers: each and
> every element in { } is a natural numbers, and all natural numbers in { } are
> "connected" (there are no gaps between them).

The emptyset is very handy. I am reminded of a story my dad told me.
His mom sent him to the store to get some empty boxes. He was perplexed
when the store owner asked "Empty boxes of what?"

William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
> On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:

> > The sequence does not stretch to 0, therefore it ends before 0
> Nope. The fact that there is a bound on the elements does not mean there is a last element. There is no end.

Either the bound is the end, or the end is even before the bound.
But there is no last element.
Don't confuse these two different notions.

Regards, WM

FromTheRafters schrieb am Mittwoch, 11. Januar 2023 um 20:03:00 UTC+1:

> If a FISON end is n then the associated endsegment beginning is n+1

Unfortunately I have called {1, 2, 3, ..., n} the FISON of n and {n, n+1, n+2, ...} the endsegment of n. It would be confusing to change this now.

Regards, WM

On 1/11/2023 4:30 PM, WM wrote:
> William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
>> On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:
>
>>> The sequence does not stretch to 0, therefore it ends before 0
>> Nope. The fact that there is a bound on the elements does not mean there is a last element. There is no end.
>
> Either the bound is the end, or the end is even before the bound.
> But there is no last element.
> Don't confuse these two different notions.
>
> Regards, WM

"notions" are a whim, a mental image, a belief, opinion.

Bound Ants
Before the Bound Ants
No Last Element Ants
Non-Existing Notional Ants
Ants that Notionalize

On 1/11/2023 4:33 PM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 11. Januar 2023 um 20:03:00 UTC+1:
>
>> If a FISON end is n then the associated endsegment beginning is n+1
>
> Unfortunately I have called {1, 2, 3, ..., n} the FISON of n and {n, n+1, n+2, ...} the endsegment of n. It would be confusing to change this now.
>
> Regards, WM

it is unimportant.

Sergi o laid this down on his screen :
> On 1/11/2023 4:30 PM, WM wrote:
>> William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
>>> On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:
>>
>>>> The sequence does not stretch to 0, therefore it ends before 0
>>> Nope. The fact that there is a bound on the elements does not mean there
>>> is a last element. There is no end.
>>
>> Either the bound is the end, or the end is even before the bound.
>> But there is no last element.
>> Don't confuse these two different notions.
>>
>> Regards, WM
>
>
> "notions" are a whim, a mental image, a belief, opinion.

http://settheory.net/foundations/theories

======================================================
Notions and objects
Each theory has its own list of notions (usually designated by common
names), formally serving as the kinds of variables it can use ; each
model interprets each notion as a set that is the common range of all
variables of this kind. For example, Euclidean geometry has the notions
of «point», «straight line», «circle» and more, and is usually
expressed using a different style of variable symbol for each. The
objects of a theory in a model, are all possible values of its
variables of all kinds (the elements of all its notions) in this
model.=======================================================

WM brought next idea :
> FromTheRafters schrieb am Mittwoch, 11. Januar 2023 um 20:03:00 UTC+1:
>
>> If a FISON end is n then the associated endsegment beginning is n+1
>
> Unfortunately I have called {1, 2, 3, ..., n} the FISON of n and {n, n+1,
> n+2, ...} the endsegment of n. It would be confusing to change this now.

So n is in the intersection of the FISON and the endsegment? Crazy man.

Why not use the non-negative integers and established truisms like
almost everybody else does?

FromTheRafters wrote:
> WM brought next idea :
>> FromTheRafters schrieb am Mittwoch, 11. Januar 2023 um 20:03:00 UTC+1:
>>
>>> If a FISON end is n then the associated endsegment beginning is n+1
>>
>> Unfortunately I have called {1, 2, 3, ..., n} the FISON of n and {n,
>> n+1, n+2, ...} the endsegment of n. It would be confusing to change
>> this now.
>
> So n is in the intersection of the FISON and the endsegment? Crazy man.
>
> Why not use the non-negative integers and established truisms like
> almost everybody else does?

He cannot miss a single occasion to obfuscate.

I suggest Müeckenheim's epitaph to be:

Wir müssen verschleiern.
Wir werden verschleiern.

(We must obfuscate.
We shall obfuscate.)

I also wish this tomb to be fulled by this trash guy and closed
forever as soon as possible for the sake of German Academy.

On 1/10/2023 5:43 PM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 10. Januar 2023 um 22:17:11 UTC+1:
>> On 1/10/2023 2:28 PM, WM wrote:

>>> According to Cantor ℕ fits between 0 and ω.
>>
>> Yes.
>> What we mean by 'ω' is the first infinite ordinal.
>> Each element n ∈ ℕ is a finite ordinal.
>> In the ordinals, 0 ≤ₒ n <ₒ w
>> ℕ fits between 0 and ω
>>
>>> So it ends before ω.
>>
>> No.
>
> It ends not after ω.

ℕ does not end.

You (WM) imagine an end of ℕ somewhere among
the "dark" numbers in ℕ.
There are no "dark" numbers in ℕ
and no element of ℕ is the end of ℕ

You have said that that "contradicts logic".
It doesn't,
it only contradicts ℕ being a FISON.

I see two ways to define the set ℕ of naturalsm
building up sequences from 0 and
trimming off excess numbers.

ℕ is ⋃𝓕𝓲𝓷
the smallest set superset of each FISON

ℕ is ⋂𝓘𝓷𝓭
the largest set subset of
each inductive subset 𝑆 of 𝑀
for any inductive 𝑀

You (WM) imagine "dark" numbers
in ⋂𝓘𝓷𝓭 but not in ⋃𝓕𝓲𝓷

However,
⋃𝓕𝓲𝓷 and ⋂𝓘𝓷𝓭 are the same set.
There are no "dark" numbers.

| ⋂𝓘𝓷𝓭 is inductive.
| For each inductive 𝐻, ⋂𝓘𝓷𝓭 ⊆ 𝐻
| | ⋃𝓕𝓲𝓷 is inductive.
| For each inductive 𝐻, ⋂𝓘𝓷𝓭 ⊆ 𝐻
| | ⋂𝓘𝓷𝓭 ⊆ ⋃𝓕𝓲𝓷
| | No _first_ j ∈ 𝐹 ∈ 𝓕𝓲𝓷 exists
| such that j ∉ ⋂𝓘𝓷𝓭 ∧ j⁻⁻ ∈ ⋂𝓘𝓷𝓭
| because ⋂𝓘𝓷𝓭 is inductive
| ∀i ∈ ⋂𝓘𝓷𝓭: i⁺⁺ ∈ ⋂𝓘𝓷𝓭
| | No k ∈ 𝐹 ∈ 𝓕𝓲𝓷 exists
| such that k ∉ ⋂𝓘𝓷𝓭
| | ⋃𝓕𝓲𝓷 ⊆ ⋂𝓘𝓷𝓭
| | ⋂𝓘𝓷𝓭 = ⋃𝓕𝓲𝓷

I've trimmed drastically for the sake of readability.
If you want more details, ask.

>> Yes,
>> ℕ fits between 0 and ω
>> and,
>> if an element of ℕ was its end,
>> then that end would be before ω
>
> There is no elements its end
> but a bunch of infinitely many
> indistinguishable elements.

No, that's not ℕ

It's not-possibly-false that that's not ℕ

----
There are these sequences of claims which
we know are all-not-first-possibly-false, and
from which we get our knowledge of elements
of ℕ, of end segments and their intersection,
and so on.

We depend upon the not-possibly-false-hood of
the descriptions of what we're reasoning about,
in order for the sequences to be all-not-first-
-possibly-false, and therefore all-not-possibly-
-false, and therefore knowledge.

If the descriptions are NOT _not-possibly-false_
the sequence is NOT _all-not-first-possibly-false_
and everything falls apart.

We do not apply the all-not-first-possibly-false
sequence method outside of conditions in which
the descriptions of what we're talking about are
not-possibly-false.

We will never apply the Pythagorean theorem to
a "right triangle" which _might not_ have
a right angle or _might not_ be a triangle.
Someone presenting _something else_ maybe
a 3:4:6 triangle, as a counter-example has
apparently not understood what is being claimed.

FromTheRafters <FTR@nomail.afraid.org> writes:
> Fritz Feldhase explained :
>> Why is { } a "finite initial segement of natural numbers"? Well,
>> it's certainly a finite set. Moreover it's a segement of natural
>> numbers: each and every element in { } is a natural numbers, and all
>> natural numbers in { } are "connected" (there are no gaps between
>> them).
>
> The emptyset is very handy. I am reminded of a story my dad told
> me. His mom sent him to the store to get some empty boxes. He was
> perplexed when the store owner asked "Empty boxes of what?"

I went to the cafe yesterday and asked for a coffee with no milk.
Staff said "Sorry, sir, we're out of milk", so I said "OK, coffee
with no cream is fine".

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have
gained some understanding of the world in which we live. As such, we can cast
aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

On Thursday, January 12, 2023 at 12:43:27 AM UTC+1, Jim Burns wrote:

> [...]
>
> I've trimmed drastically for the sake of readability.
> If you want more details, ask.

I'm quite sure that he will not miss this opportunity!

On Wednesday, January 11, 2023 at 6:30:13 PM UTC-4, WM wrote:
> William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
> > On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:
>
> > > The sequence does not stretch to 0, therefore it ends before 0
> > Nope. The fact that there is a bound on the elements does not mean there is a last element. There is no end.
> Either the bound is the end, or the end is even before the bound.
Nope. There is a bound, but there is no end.
> But there is no last element.
If there is no last element there is no end..

onsdag 11 januari 2023 kl. 17:27:58 UTC+1 skrev WM:
> William schrieb am Mittwoch, 11. Januar 2023 um 15:50:09 UTC+1:
> > On Wednesday, January 11, 2023 at 7:48:34 AM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 10. Januar 2023 um 23:48:11 UTC+1:
> > > > On Tuesday, January 10, 2023 at 6:36:03 PM UTC-4, WM wrote:
> > > >
> > > > > The sequence (1/n) ends
> > > >
> > > > Nope. It does not have an end.
> > > It streches to the negative axis?
> > Nope. Every element is greater than 0. However, there is no last element. There is no end.
> The sequence does not stretch to 0, therefore it ends before 0 but there is no last element. How can these two diverging properties be reconciled?
>
> The only way to describe an end without identifying a last element is the assumption of dark elements.
>
> Regards, WM
There is no end to a sequence you retard

William schrieb am Donnerstag, 12. Januar 2023 um 03:06:26 UTC+1:
> On Wednesday, January 11, 2023 at 6:30:13 PM UTC-4, WM wrote:
> > William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
> > > On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:
> >
> > > > The sequence does not stretch to 0, therefore it ends before 0
> > > Nope. The fact that there is a bound on the elements does not mean there is a last element. There is no end.
> > Either the bound is the end, or the end is even before the bound.
> Nope. There is a bound, but there is no end.
> > But there is no last element.
> If there is no last element there is no end.

ω is Cantor's invention. We can get rid of dark numbers in that realm by throwing it out of the club of ordinals. But the sequence (1/n) converges to 0 and does not cross it. Therefore it ends there or before.
But there is no last term of the sequence visible when we move from 1 to 0, passing *all* terms.
Therefore not all terms are visible.

Regards, WM

Jim Burns schrieb am Donnerstag, 12. Januar 2023 um 00:43:27 UTC+1:

> There are no "dark" numbers in ℕ
> and no element of ℕ is the end of ℕ

ω is Cantor's invention. We can get rid of dark numbers in that realm by throwing it out of the club of ordinals. But the sequence (1/n) converges to 0 and does not cross it. Therefore it ends there or before.
But there is no last term of the sequence visible when we move from 1 to 0, passing *all* terms.
Therefore not all terms are visible.

Regards, WM

torsdag 12 januari 2023 kl. 09:58:35 UTC+1 skrev WM:
> William schrieb am Donnerstag, 12. Januar 2023 um 03:06:26 UTC+1:
> > On Wednesday, January 11, 2023 at 6:30:13 PM UTC-4, WM wrote:
> > > William schrieb am Mittwoch, 11. Januar 2023 um 19:20:38 UTC+1:
> > > > On Wednesday, January 11, 2023 at 12:27:58 PM UTC-4, WM wrote:
> > >
> > > > > The sequence does not stretch to 0, therefore it ends before 0
> > > > Nope. The fact that there is a bound on the elements does not mean there is a last element. There is no end.
> > > Either the bound is the end, or the end is even before the bound.
> > Nope. There is a bound, but there is no end.
> > > But there is no last element.
> > If there is no last element there is no end.
> ω is Cantor's invention. We can get rid of dark numbers in that realm by throwing it out of the club of ordinals. But the sequence (1/n) converges to 0 and does not cross it. Therefore it ends there or before.
> But there is no last term of the sequence visible when we move from 1 to 0, passing *all* terms.

>
> Regards, WM

Or you idiot, IT DOES NOT END!

> Therefore not all terms are visible.

Not at all retard, it does not follow

WM has brought this to us :
> Jim Burns schrieb am Donnerstag, 12. Januar 2023 um 00:43:27 UTC+1:
>
>> There are no "dark" numbers in ℕ
>> and no element of ℕ is the end of ℕ
>
> ω is Cantor's invention. We can get rid of dark numbers in that realm by
> throwing it out of the club of ordinals.

Ah, your Axiom of Convenience again. Sorry, but ordinals are "defined"
and the little omega symbol represents the 'next' ordinal number after
the set of ordered naturals.

Phil Carmody brought next idea :
> FromTheRafters <FTR@nomail.afraid.org> writes:
>> Fritz Feldhase explained :
>>> Why is { } a "finite initial segement of natural numbers"? Well,
>>> it's certainly a finite set. Moreover it's a segement of natural
>>> numbers: each and every element in { } is a natural numbers, and all
>>> natural numbers in { } are "connected" (there are no gaps between
>>> them).
>>
>> The emptyset is very handy. I am reminded of a story my dad told
>> me. His mom sent him to the store to get some empty boxes. He was
>> perplexed when the store owner asked "Empty boxes of what?"
>
> I went to the cafe yesterday and asked for a coffee with no milk.
> Staff said "Sorry, sir, we're out of milk", so I said "OK, coffee
> with no cream is fine".
>
> Phil

I like it. :oD