Once upon a time there were six sisters who got lost in the dark forest. There an evil sorcerer found them and decided, first to fatten them and then to eat them.

When the sisters recognized their fate they craved mercy. The evil sorcerer, after short reflection, put them a task. Every sister should name a sequence of sets. The evil sorcerer promised to release those girls who, at the end, could show him a set with ℵ₀ elements. (Of course he knew that none of them had ever heard of ℵ₀ – as I said he was an evil sorcerer.)

The two eldest, Anna and Bertha, began. They knew the even numbers already. So they simply used the sequence of finite initial segments of even numbers {2, 4, 6, ..., 2n}. Since each one should have her own sequence, Bertha took all numbers that were larger than n and Anna took the rest. So the sisters believed to share just and fair.

n A(n) B(n)
1 { } {2}
2 {2} {4}
3 {2} {4, 6}
4 {2, 4} {6, 8}
5 {2, 4} {6, 8, 10}
6 {2, 4, 6} {8, 10, 12}
... ... ... .

Alas, this was fateful for Bertha. The evil sorcerer decided that Bertha's sequence had the empty set as its limit set, because for every even number there is a natural number, a step n, where it enters Anna's set.

Christa and Dora, who had observed Bertha's misfortune, were cleverer. They chose the sequences

n C(n) D(n)
1 { } {1}
2 {2} {1}
3 {2} {1, 3}
4 {2, 4} {1, 3}
5 {2, 4} {1, 3, 5}
6 {2, 4, 6} {1, 3, 5}
... ... ... .

And the evil sorcerer had to confess that both had solved their task.

Now Edda and Frida remained. Frida had already recognized the whole lot, but she looked for a way to break the spell of poor Bertha. And when Edda started, Frida took a really sophisticated choice:

n E(n) F(n)
1 { } {1}
2 {2} {1}
3 {2} {1, 2}
4 {2, 4} {1, 2}
5 {2, 4} {1, 2, 3}
6 {2, 4, 6} {1, 2, 3}
... ... ... .

Here identical indices should not imply equal elements but only distinguish free places. And although the evil sorcerer threatened her harshly, she did not reveal whether her squares should be filled with Bertha's or Dora's elements. This was too much for the evil sorcerer. His brain overheated and finally burst. And all six sisters were free and could run home.

Frida shows that the results of set theory depend on arbitrarily selected notation. That must never be accepted in science and not even in fairy tales..

[W. Mückenheim: "Die Geschichte des Unendlichen", 7th ed., Maro, Augsburg (2011) p. 113f]

On 12/9/2021 1:24 AM, WM wrote:
> Once upon a time there were six sisters who got lost in the dark forest. There an evil sorcerer found them and decided, first to fatten them and then to eat them.
[...]

Are you loaded or something?

WM <wolfgang.mueckenheim@hs-augsburg.de> writes:

> ... Every sister should name a sequence of sets. The evil sorcerer
> promised to release those girls who, at the end, could show him a set
> with ℵ₀ elements.

None of the seems to have read your book. All they need is the
definition of ℵ₀, and the rest is in your book.

--
Ben.

On Thursday, December 9, 2021 at 5:25:25 AM UTC-5, Chris M. Thomasson wrote:
> On 12/9/2021 1:24 AM, WM wrote:
> > Once upon a time there were six sisters who got lost in the dark forest.. There an evil sorcerer found them and decided, first to fatten them and then to eat them.
> [...]
>
> Are you loaded or something?

Loaded or not, he's got the sixes. I hope one of those sisters is old enough for me.
Aleph has another place I believe.
Down in the world of finite set theory we do have need of something like the aleph.
It is buried in the radix theory.
When we utter things like:
666 = 6 x 10 ^ 2 + 6 x 10 ^ 1 + 6 x 10 ^ 0
this value '10' and its interpretation is ambiguous.
It is rather an aleph, and without the digits preceding it not much can be said about the above line, for if I were to issue the digits thus:
0 1 2 3 5 6 7 8 9 A
and here I have induced an 'A' for Aleph; a value whose use within the ordinary number theory will not occur, yet whose interpretation is unmistakably that of '10', which is a two digit number. Here lays the ambiguity: if it is possible to explain these multi-digit numbers then shouldn't it be done without two digit numbers? Here we go again, bringing mathematicians into the digital universe. If you've read carefully above here you'll see 410 in your ordinary language.

Of course the whole exponential representation is a farce at this level. Its truth is out a level. The digits themselves are carrying qualities which allow this exponential interpretation, and yet clearly the values preceded it. This issue of numerical construction is a sticky one so it is understandable how mathematicians would like to leave it behind. And so they do. How very careful of them.

on 12/9/2021, Chris M. Thomasson supposed :
> On 12/9/2021 1:24 AM, WM wrote:
>> Once upon a time there were six sisters who got lost in the dark forest.
>> There an evil sorcerer found them and decided, first to fatten them and
>> then to eat them.
> [...]
>
> Are you loaded or something?

Some guys just like the girls who have bigger sets. I'm more of a
foundations man myself.

On 12/9/2021 3:24 AM, WM wrote:
> Once upon a time there were six sisters who got lost in the dark forest. There an evil sorcerer found them and decided, first to fatten them and then to eat them.
>
> When the sisters recognized their fate they craved mercy. The evil sorcerer, after short reflection, put them a task. Every sister should name a sequence of sets. The evil sorcerer promised to release those girls who, at the end, could show him a set with ℵ₀ elements. (Of course he knew that none of them had ever heard of ℵ₀ – as I said he was an evil sorcerer.)
>
> The two eldest, Anna and Bertha, began. They knew the even numbers already. So they simply used the sequence of finite initial segments of even numbers {2, 4, 6, ..., 2n}. Since each one should have her own sequence, Bertha took all numbers that were larger than n and Anna took the rest. So the sisters believed to share just and fair.
>
> n A(n) B(n)
> 1 { } {2}
> 2 {2} {4}
> 3 {2} {4, 6}
> 4 {2, 4} {6, 8}
> 5 {2, 4} {6, 8, 10}
> 6 {2, 4, 6} {8, 10, 12}
> ... ... ... .
>

so Anna and Bertha never finished, in fact they are STILL listing sequences, as it takes them a finite amount of time to do so ( the three dots ...
means to infinity, no stopping)

> Alas, this was fateful for Bertha. The evil sorcerer decided that Bertha's sequence had the empty set as its limit set, because for every even number there is a natural number, a step n, where it enters Anna's set.

evil sorcerer is a twice dumbo, as they are not finished yet, and both gain more elements per double step

>
> Christa and Dora, who had observed Bertha's misfortune, were cleverer. They chose the sequences
>
> n C(n) D(n)
> 1 { } {1}
> 2 {2} {1}
> 3 {2} {1, 3}
> 4 {2, 4} {1, 3}
> 5 {2, 4} {1, 3, 5}
> 6 {2, 4, 6} {1, 3, 5}
> ... ... ... .
>
> And the evil sorcerer had to confess that both had solved their task.
>
> Now Edda and Frida remained. Frida had already recognized the whole lot, but she looked for a way to break the spell of poor Bertha. And when Edda started, Frida took a really sophisticated choice:
>
> n E(n) F(n)
> 1 { } {1}
> 2 {2} {1}
> 3 {2} {1, 2}
> 4 {2, 4} {1, 2}
> 5 {2, 4} {1, 2, 3}
> 6 {2, 4, 6} {1, 2, 3}
> ... ... ... .
>
> Here identical indices should not imply equal elements but only distinguish free places. And although the evil sorcerer threatened her harshly, she did not reveal whether her squares should be filled with Bertha's or Dora's elements. This was too much for the evil sorcerer. His brain overheated and finally burst. And all six sisters were free and could run home.

agree, es was a snot, and dosent know math.

>
> Frida shows that the results of set theory depend on arbitrarily selected notation.

Frida got a F in set theory.

>That must never be accepted in science and not even in fairy tales.
>
> [W. Mückenheim: "Die Geschichte des Unendlichen", 7th ed., Maro, Augsburg (2011) p. 113f]
>

please add my corrections to the reference