On 9/9/2022 2:19 PM, Fritz Feldhase wrote:
> On Friday, September 9, 2022 at 7:06:37 PM UTC+2, Jim Burns wrote:
>
>> You might have seen me and others write phrases like "ℕ the natural numbers".
>
> What a shame. You shouldn't write that since
>
>> "ℕ the natural numbers" is technically incorrect,
>
> Right.
>
>> but we write it anyway with the
>> expectation that those for whom we intend
>> our remarks will know that we know it's
>> incorrect as written and will read it as
>> a shortened version of
>> "the set ℕ of the natural numbers".
>
> If you say so.
>
> Actually, I like to be able to talk about the /natural numbers/ (i. e. the objects in IN), in contrast to the /set of natural numbers/, IN. :-P
>
> When "talking" to a crank such things may matter.

Replace those pesky worn out natural numbers with the Counting Numbers!

You can buy sets of the Counting Numbers with sticky back on my eBay, and put them on your sheeps or rocks, people etc. {the full set of Counting
Numbers is $4.95 per month, we ship you 100,000 each month, and we dont lose count of where you are either) Send money to makemerich@paypal.com. If you
want expedited, we make contract with Chuck Norris.

On 9/9/2022 3:10 PM, WM wrote:
> Jim Burns schrieb am Freitag, 9. September 2022 um 19:06:37 UTC+2:
>> On 9/9/2022 4:14 AM, WM wrote:
>
>> A FISON ends somewhere.
>> Each element in a FISON
>> is in a finite set.
>>
>> The set of things-in-FISONs doesn't end
>> anywhere.
>
> We know that no FISON is actually infinite, i.e. larger than every FISON. More is not required to know that everything in FISONs is finite.
>
>> Each element in the set of things-in-FISONs
>> is in an infinite set.
>
> Potentially infinite but not really infinite.
>
>> ∅ is next[2] after all end segments
>
> This is my last answer to you, if your stupidity is not aible to recognize your mistake.
>
> After all infinite endsegments with infinite intersections
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> the empty intersection cannot immediately follow because of
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}.
> Only one element can be lost per step. The empty set cannot follow immediately upon infinite sets.

....back to the mush. No Math here.

>
> Regards, WM
>

On 9/9/2022 4:10 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 9. September 2022 um 19:06:37 UTC+2:

>>>> Yes,
>>>> for each FISON 𝐹,
>>>> the set 𝐸 of things-in-FISONs-following-𝐹
>>>> is infinite.
>>>
>>> Not things-in-FISONs.

>> Each element in a FISON
>> is in a finite set.
>>
>> The set of things-in-FISONs doesn't end
>> anywhere.
>
> We know that no FISON is actually infinite,
> i.e. larger than every FISON.
> More is not required to know that
> everything in FISONs is finite.

The set ⋃𝓕 of things-in-FISONs is infinite.

>> Each element in the set of things-in-FISONs
>> is in an infinite set.
>
> Potentially infinite but
> not really infinite.

Each thing-in-a-FISON n
ends a FISON 𝐹ₙ
and
has a successor n+1
which
ends a FISON 𝐹ₙ⊕⟨n+1⟩
and, thus,
is also a thing-in-a-FISON.

If n is a thing-in-a-FISON then n ∈ ⋃𝓕
If n ∈ ⋃𝓕 then n is a thing-in-a-FISON

∀n ∈ ⋃𝓕, n+1 ∈ ⋃𝓕

Suppose that ⋃𝓕 is "not really infinite".

If p,q ∈ ⋃𝓕
then
s = p+q, s ∈ ⋃𝓕
n = (s-1)(s-2)/2+p, n ∈ ⋃𝓕

If n ∈ ⋃𝓕
then
s = ⌈(((8n+1)¹ᐟ²+1)/2)⌉, s ∈ ⋃𝓕
p = n-(s-1)(s-2)/2, p ∈ ⋃𝓕
q = s-p, q ∈ ⋃𝓕

Suppose that ⋃𝓕 is "not really infinite".
So what?

>> ∅ is next[2] after all end segments
>
> This is my last answer to you,
> if your stupidity is not aible to
> recognize your mistake.

> Only one element can be lost per step.
> The empty set cannot follow immediately
> upon infinite sets.

There is no set 𝐵 closer than ∅
What would make 𝐵 closer?

An element i not-in ⋃𝓕?
Why?
i is not in any end segment.

An element i in ⋃𝓕?
But there's an infinite end segment E(i+1)
with i not in it.
𝐵 with i isn't after E(i+1)

Ben Bacarisse expressed precisely :
> Jim Burns <james.g.burns@att.net> writes:
>
>> I don't know if this is current WM-dogma,
>> but, at one time, WM did not accept
>> that the empty set ∅ existed.
>
> Well, it's right there in his textbook.

When something 'gets in his way' he makes it non-existent by his Axiom
Of Because I Say So.

After serious thinking Sergio wrote :
> On 9/9/2022 2:19 PM, Fritz Feldhase wrote:
>> On Friday, September 9, 2022 at 7:06:37 PM UTC+2, Jim Burns wrote:
>>
>>> You might have seen me and others write phrases like "ℕ the natural
>>> numbers".
>>
>> What a shame. You shouldn't write that since
>>
>>> "ℕ the natural numbers" is technically incorrect,
>>
>> Right.
>>
>>> but we write it anyway with the
>>> expectation that those for whom we intend
>>> our remarks will know that we know it's
>>> incorrect as written and will read it as
>>> a shortened version of
>>> "the set ℕ of the natural numbers".
>>
>> If you say so.
>>
>> Actually, I like to be able to talk about the /natural numbers/ (i. e. the
>> objects in IN), in contrast to the /set of natural numbers/, IN. :-P
>>
>> When "talking" to a crank such things may matter.
>
>
>
> Replace those pesky worn out natural numbers with the Counting
> Numbers!
>
> You can buy sets of the Counting Numbers with sticky back on my eBay, and put
> them on your sheeps or rocks, people etc. {the full set of Counting Numbers
> is $4.95 per month, we ship you 100,000 each month, and we dont lose count of
> where you are either) Send money to makemerich@paypal.com. If you want
> expedited, we make contract with Chuck Norris.

In a pinch, one can use certain subsets of integers instead. Of course,
you would have to have room to store the unused portions.

WM presented the following explanation :
> Jim Burns schrieb am Freitag, 9. September 2022 um 19:06:37 UTC+2:
>> On 9/9/2022 4:14 AM, WM wrote:
>
>> A FISON ends somewhere.
>> Each element in a FISON
>> is in a finite set.
>>
>> The set of things-in-FISONs doesn't end
>> anywhere.
>
> We know that no FISON is actually infinite,

All natural numbers less than or equal to n cannot be anything else but
finite. You suffer from delusion monotony.

On 9/9/2022 5:40 PM, FromTheRafters wrote:
> WM presented the following explanation :
>> Jim Burns schrieb am Freitag, 9. September 2022 um 19:06:37 UTC+2:
>>> On 9/9/2022 4:14 AM, WM wrote:
>>
>>> A FISON ends somewhere. Each element in a FISON is in a finite set.
>>> The set of things-in-FISONs doesn't end anywhere.
>>
>> We know that no FISON is actually infinite,
>
> All natural numbers less than or equal to n cannot be anything else but finite. You suffer from delusion monotony.

good one!

On Friday, September 9, 2022 at 10:10:14 PM UTC+2, WM wrote:
>
> We know that no FISON is actually infinite

Yeah, since each and every FISON is actually finite, by definition.

> > Each element in [the set of natural numbers] is in an infinite set.

Indeed! Since the set of natural numbers IN is an infinite set (and each and every natural number is in IN, by definition).

That's a provable fact in the context of set theory.

> Potentially infinite but not really infinite.

Psychotic nonsense uttered by a crank.

> This is my last answer to you, if your stupidity is not able to recognize your mistake.

Once more, again?

> The empty set cannot follow immediately upon [all endsegments].

The set {E_n : n e IN} u {{}} is certainly ordered the following way (by the subset relation):

E_1 > E_2 > E_3 > ... > {}.

Moreover, there is _no_ nonempty set X such that

E_1 > E_2 > E_3 > ... > X > {}.

And now fuck off, you *silly* asshole!

On Friday, September 9, 2022 at 10:10:14 PM UTC+2, WM wrote:
>
> We know that no FISON is actually infinite

Yeah, since each and every FISON is actually finite, by definition.

> > Each element in [the set of natural numbers] is in an infinite set.

Indeed! Since the set of natural numbers IN is an infinite set (and each and every element in IN is in IN).

That's a provable fact in the context of set theory.

> Potentially infinite but not really infinite.

Psychotic nonsense uttered by a crank.

> This is my last answer to you, if your stupidity is not able to recognize your mistake.

Once more, again?

> The empty set cannot follow immediately upon [all endsegments].

The set {E_n : n e IN} u {{}} is certainly ordered the following way (by the subset relation):

E_1 > E_2 > E_3 > ... > {}.

Moreover, there is _no_ nonempty set X such that

E_1 > E_2 > E_3 > ... > X > {}.

And now fuck off, you *silly* asshole!

On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:

> All natural numbers less than or equal to n cannot be anything else but
> finite.

Should read: "The set of all natural numbers less than or equal to n cannot be anything else but finite."

On Friday, September 9, 2022 at 10:16:14 PM UTC+2, WM wrote:
> Gus Gassmann schrieb am Freitag, 9. September 2022 um 20:10:30 UTC+2:
> > On Friday, 9 September 2022 at 14:33:42 UTC-3, WM wrote:
> > [...]
> > > Limits do not matter for the question whether it is possible to enumerate all fractions by natural numbers, i.e., before any limit.
> > If you insist on a stepwise process,
> I do not insist. Cantorhas claimed that he could enumerate all fractions. The indices are a stepwise well-ordered set 1, 2, 3, ... .
> > then there is a limit to consider.
> What happens in the limit is not what Cantor has claimed.
>
> "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist, immer auch noch auf viele andere Weisen geschehen kann), so möge für das Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p. 119]
>
> No limit.
>
> "Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollständig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
>
> No limit.
> > If you allow for a completed process, then there is no question of how to transform
> > XOOO...
> > XOOO...
> > XOOO...
> > XOOO...
> > ...
> > into
> >
> > XXXX...
> > XXXX...
> > XXXX...
> > XXXX...
> > ...
> >
> > since both matrices exist simultaneously. (Obviously.) What you want is to have both a stepwise process, and no limit.
> That is what Cantor claimed.
>
> > That is not possible,
>
> Of course not. Nevertheless it is what Cantor claimed and what his disciples usually believe. His claim cannot be realized. Dark numbers prevent it! Nice that you have grasped it after all.
>
> Regards, WM

OK, Mücke. I retract everything I have ever said.
You have convinced me: Dark numbers prevent it!

Now our problem is to convince the rest of the world.

To that end we need to make several things clear:
1. What exactly are dark numbers?
2. Which property of dark numbers prevents what exactly?
3. What is a matrix? What is an infinite matrix?
4. What are the admissible operation on an infinite matrix?

So lets get to work! I will help you; and the rest of the audience
is invited to contribute.

Note to Ben Bacarisse:
Infinite matrices are not defined in Mücke's Bestseller.
In fact, matrices are not really defined.
Moreover, the term 'enumerable' (abzählbar, Abzählbarkeit) doesn't occur in
the book.
Of course, we can construct such an object using information
from the book. In any case, it might be useful to agree on the
nature of the object under discussion.

On Friday, September 9, 2022 at 3:08:07 PM UTC+2, WM wrote:
> Ben Bacarisse schrieb am Freitag, 9. September 2022 um 02:51:25 UTC+2:
> > WM <mont...@t-online.de> writes:
> > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> > Unendlichen" at Hochschule Augsburg.)
> > > Ben Bacarisse schrieb am Donnerstag, 8. September 2022 um 02:17:23 UTC+2:
> > >> WM <askas...@gmail.com> writes:
> > >
> > >> We /could/ try to count number of Xs and Os but your textbook does not
> > >> even say how that can be done
> > >
> > > This logic is too simple to be covered in a book for universities. It
> > > simply states that exchanging X and O will never change the number of
> > > X's and O's. Therefore these numbers need not be counted.
> > Interesting! So in WMaths the "number of Xs" means something? What is
> > that number and how is it defined? Are the "number of Xs" and the
> > "number of Os" equal, and how do you define equality for such numbers?
> In classical mathematics, there is no alephs. But we can use classical maths to estimate, according to an argument frequently used in number theory, that in every definable step more than 1000 O's remain. For unsure and doubting readers, we could also say more than 10^10^1000 O's are remaining whenever an index settles at its finite position.
> > But I used WMaths to show which positions have Os at each step.
> You agreed that no O is deleted during the whole process. That's what is important! Do you wish to withdraw it?
>
> Regards, WM

And, of course, I don't want to interfere in any way with your interchange
with Ben B.
But if you want to convince others, as you have convinced me, that your insights
are sound, it would help to clarify one or two details.

Fritz Feldhase pretended :
> On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:
>
>> All natural numbers less than or equal to n cannot be anything else but
>> finite.
>
> Should read: "The set of all natural numbers less than or equal to n cannot
> be anything else but finite."

Might also read: The sequence of all natural numbers less than or equal
to n cannot be anything else but finite.

On Saturday, September 10, 2022 at 1:12:57 PM UTC+2, FromTheRafters wrote:
> Fritz Feldhase pretended :
> > On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:

WM was talking about certein SETs:

> > We know that no FISON is actually infinite,"

Your comment:

> All natural numbers less than or equal to n cannot be anything else but finite.

Hence my hint:

Should read: "The set of all natural numbers less than or equal to n cannot be anything else but finite."

EOD

Fritz Feldhase submitted this idea :
> On Saturday, September 10, 2022 at 1:12:57 PM UTC+2, FromTheRafters wrote:
>> Fritz Feldhase pretended :
>>> On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:
>
> WM was talking about certein SETs:

Was he? A segment is a portion of a sequence. The 'S' in FISON stands
for segment and 'endsegment' has the whole word in it.

>>> We know that no FISON is actually infinite,"
>
> Your comment:
>
>> All natural numbers less than or equal to n cannot be anything else but
>> finite.
>
> Hence my hint:
>
> Should read: "The set of all natural numbers less than or equal to n cannot
> be anything else but finite."
>
> EOD

Then shut up already.

On 9/9/2022 11:17 PM, Fritz Feldhase wrote:
> On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:
>
>> All natural numbers less than or equal to n cannot be anything else but
>> finite.
>
> Should read: "The set of all natural numbers less than or equal to n cannot be anything else but finite."

more concisely: "A set of natural numbers less than n is finite."

on 9/10/2022, Sergio supposed :
> On 9/9/2022 11:17 PM, Fritz Feldhase wrote:
>> On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:
>>
>>> All natural numbers less than or equal to n cannot be anything else but
>>> finite.
>>
>> Should read: "The set of all natural numbers less than or equal to n cannot
>> be anything else but finite."
>
>
>
> more concisely: "A set of natural numbers less than n is finite."

The initial 'segment' of the 'sequence' of well-ordered natural numbers
in their natural order is defined differently for naturals with zero as
opposed to naturals without zero. Both are finite though.

Jim Burns schrieb am Freitag, 9. September 2022 um 22:30:02 UTC+2:

> I don't know if this is current WM-dogma,
> but, at one time, WM did not accept
> that the empty set ∅ existed.

I am not the only one who is in doubt.

Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [B. Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]

Gottlob Frege shared his opinion: "If, according to our previous use of the word, a class consists of things, is a collection, a collective union of them, then it must disappear when these things disappear. If we burn down all the trees of a forest, then we burn down the forest. Thus an empty class cannot exist." [G. Frege: "Kleine Schriften", I. Agelelli (ed.), 2nd ed., Olms, Hildesheim (1990) p. 195]

Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O; P  O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146]

And even Ernst Zermelo who made the "Axiom II. There is an (improper) set, the 'null-set' 0 which does not contain any elements" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same author himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]

I think it is only the expression for nothing. Nothing is named the empty set.

Regards, WM

On 9/10/2022 10:23 AM, WM wrote:
> Jim Burns schrieb am Freitag, 9. September 2022 um 22:30:02 UTC+2:
>
>> I don't know if this is current WM-dogma,
>> but, at one time, WM did not accept
>> that the empty set ∅ existed.
>
> I am not the only one who is in doubt.

and you do doubt most mathematics,

only one of your quotes below supports your stand.

>
> Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [B. Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]

Russell does not deny the empty set.

>
> Gottlob Frege shared his opinion: "If, according to our previous use of the word, a class consists of things, is a collection, a collective union of them, then it must disappear when these things disappear. If we burn down all the trees of a forest, then we burn down the forest. Thus an empty class cannot exist." [G. Frege: "Kleine Schriften", I. Agelelli (ed.), 2nd ed., Olms, Hildesheim (1990) p. 195]

too simplistic analogy

>
> Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O; P  O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146]

Cantor approved of the empty set and used it.

>
> And even Ernst Zermelo who made the "Axiom II. There is an (improper) set, the 'null-set' 0 which does not contain any elements" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same author himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]

Zermelo approved and used the empty set.

>
> I think it is only the expression for nothing. Nothing is named the empty set.

Wrong again, it is a set without elements.

>
> Regards, WM

On Saturday, September 10, 2022 at 5:23:06 PM UTC+2, WM wrote:

> Gottlob Frege shared his opinion

No, he didn't, you lying piece of shit!

Ben Bacarisse schrieb am Freitag, 9. September 2022 um 23:06:26 UTC+2:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > It is what we have been discussing in extensio. It is nonsense to
> > claim the indexing of all fractions if most will be remaining not
> > indexed forever.
> My argument was entirely within WMaths. It used a potentially infinite
> bijection between potentially infinite sets to define a potentially
> infinite sequence of potentially infinite functions.

There is no problem. But I have been analyzing Cantor's completed infinity with a complete matrix of fractions.

> It then used the
> appropriate limit as defined in your book to obtain the result. Since
> everything is potential, the only reasonable meaning for "the result" is
> the limit produce by the potentially infinite sequence of swaps.

Cantor has explicitly denied the potential infinity.

"Nevertheless the transfinite cannot be considered a subsection of what is usually called 'potentially infinite'. Because the latter is not (like every individual transfinite and in general everything due to an 'idea divina') determined in itself, fixed, and unchangeable, but a finite in the process of change, having in each of its current states a finite size; like, for instance, the temporal duration since the beginning of the world, which, when measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]

> > I prefer to stay with endsegments.
> Yet you agree that infinite non-zero points "vanish" in the limit,

In the limit of real functions which do not need a mechanism to get rid of their load there is no problem.

> > There the direct drop from infinity
> > to zero is impossible by the very definition:
> >
> > ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> >
> > I assume that
> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > holds for all natural numbers.
> > Maybe you are not agreeing?
> You've been saying this for years with no one disagreeing.

Jim Birns has claimed that the empty set is next to the infinite. That means he is disagreeing.

>
> But, since you prefer to re-phrase my limit example in terms of
> end segments, here's what you didn't know how to write:
>
> Given e_n(i) = [i >= n], then lim_{n->oo} e_n = z where z(i) = 0.

If the limit is zero, then before the limit there are finite endsegments. Of course not findable, they are dark.

> > The argument proves dark numbers, not individually definable but only
> > complete in whatever limit.
> If you were intellectually curious (and courageous), you'd try to
> understand my argument -- the one you won't look

I know it! The limit! I have accepted that infinitely many elements which are present in every endsegmnet will disappear in the limit. Fine. Alas, if you need the limit then they cannot disappear individually - like every definable element. They are dark.

> it is entirely in WMaths and there are no dark numbers in WMaths.

There is no actual infinity in my book. Nevertheless, here I talk about Cantor.

> > handles infinitely many undefinable elements?
> The limits in WMaths don't do that.

There are no set limits in my book.

Regards, WM

On Saturday, 10 September 2022 at 12:23:06 UTC-3, WM wrote:
[...]

> I think it is only the expression for nothing. Nothing is named the empty set.

What you call it is ultimately irrelevant. It is important that no natural number is element of the intersection over all end segments.It is equally important (and easier) that there is no natural number that is both even and odd.I call each intersection the empty set, because it does not matter how the intersection is formed. If it contains no element, then it contains no element.

On Saturday, September 10, 2022 at 3:52:29 PM UTC+2, FromTheRafters wrote:
> Fritz Feldhase wrote :
> >
> > WM was talking about certein SETs:
> >
> Was he?

YES, HE WAS, YOU FUCKING ASSHOLE FULL OF SHIT!!!

DUMB AS A BRICK TOO!

> A segment is a portion of a sequence.

Holy shit! <facepalm>

> The 'S' in FISON stands for segment and 'endsegment' has the whole word in it.

Look, you asshole, FISONs [in the pressent context] are defined as the sets in

{{k e IN : k <= n} : n e IN} [with IN = {1, 2, 3, ...}]

They are no "sequences" but just (certain) finite sets of natural number!!!

What's the matter with you internet idiots?!

Anyway, your comment:

| "All natural numbers less than or equal to n cannot be anything else but finite."

is just bullshit.

(Though it's not wrong from a set theoretic point of view, if "the natural numbers" are defined due to von Neumann. But that was't the topic here.)

You were mixing up:

Ak e IN(k <= n -> k is finite) [for any n e IN]

with

{k e IN : k <= n} is finite [for any n e IN]

> > Hence my hint:
> >
> > Should read: "The set of all natural numbers less than or equal to n cannot
> > be anything else but finite."

AND NOW FUCK OFF FOR GOOD, YOU FUCKING ASSHOLE FULL OF SHIT!!!

On 9/10/2022 10:50 AM, WM wrote:
> Ben Bacarisse schrieb am Freitag, 9. September 2022 um 23:06:26 UTC+2:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>>> It is what we have been discussing in extensio. It is nonsense to
>>> claim the indexing of all fractions if most will be remaining not
>>> indexed forever.
>> My argument was entirely within WMaths. It used a potentially infinite
>> bijection between potentially infinite sets to define a potentially
>> infinite sequence of potentially infinite functions.
>
> There is no problem. But I have been analyzing Cantor's completed infinity with a complete matrix of fractions.
>
>> It then used the
>> appropriate limit as defined in your book to obtain the result. Since
>> everything is potential, the only reasonable meaning for "the result" is
>> the limit produce by the potentially infinite sequence of swaps.
>
> Cantor has explicitly denied the potential infinity.
>
> "Nevertheless the transfinite cannot be considered a subsection of what is usually called 'potentially infinite'. Because the latter is not (like every individual transfinite and in general everything due to an 'idea divina') determined in itself, fixed, and unchangeable, but a finite in the process of change, having in each of its current states a finite size; like, for instance, the temporal duration since the beginning of the world, which, when measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]

No, in above Cantor says 'potentially infinite' is a finite in the process of change with time, always remaining finite.

> Regards, WM

On Saturday, September 10, 2022 at 5:23:06 PM UTC+2, WM wrote:
> Jim Burns schrieb am Freitag, 9. September 2022 um 22:30:02 UTC+2:
>
> > I don't know if this is current WM-dogma,
> > but, at one time, WM did not accept
> > that the empty set ∅ existed.
>
> I am not the only one who is in doubt.

NO ONE -except you- *is* in doubt, you psychotic lying asshole full of shit!