Le 20/04/2024 à 19:46, FromTheRafters a écrit :

WM used his keyboard to write :

Then that one directly before ω is not multiplied. Or it is not existing. But
what exists directly before ω?

All of the previous (finite) ordinals.

Multiplying by 2 changes the multiplied ordinal.

Regards, WM

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Le 10/05/2024 à 17:00, FromTheRafters a écrit :

WM explained :

Yes. But every n ∈ ℕ_def has ℵ₀ successors which never vanish by
counting.
They can be removed only collectively such that nothing of ℕ remains.

Wrong,

Try to remove all natural numbers individually from ℕ. Fail.

Regards, WM

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Le 14/05/2024 à 00:47, FromTheRafters a écrit :

WM expressed precisely :

Le 10/05/2024 à 17:00, FromTheRafters a écrit :

WM explained :

Yes. But every n ∈ ℕ_def has ℵ₀ successors which never vanish by >>>> counting.
They can be removed only collectively such that nothing of ℕ remains. >>>

Wrong,

Try to remove all natural numbers individually from ℕ. Fail.

Because you cannot remove them -- *SETS DO NOT CHANGE*.

But the result of subtraction can be calculated. That is understood by removing.

Regards, WM

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Le 14/05/2024 à 01:32, "Chris M. Thomasson" a écrit :

On 5/13/2024 1:21 PM, WM wrote:

Le 10/05/2024 à 17:00, FromTheRafters a écrit :

WM explained :

Yes. But every n ∈ ℕ_def has ℵ₀ successors which never vanish by >>>> counting. They can be removed only collectively such that nothing of
ℕ remains.

Wrong,

Try to remove all natural numbers individually from ℕ. Fail.

Huh? ℕ is infinite in and of itself.

Nevertheless all natural numbers can be removed collectively. But not individually. That shows a difference, resolved by numbers which can only
be treated collectively: dark numbers.

Regards, WM

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Le 14/05/2024 à 16:15, FromTheRafters a écrit :

WM expressed precisely :

Le 14/05/2024 à 00:47, FromTheRafters a écrit :

WM expressed precisely :

Le 10/05/2024 à 17:00, FromTheRafters a écrit :

WM explained :

Yes. But every n ∈ ℕ_def has ℵ₀ successors which never vanish by >>>>>> counting. They can be removed only collectively such that nothing of ℕ >>>>>> remains.

Wrong,

Try to remove all natural numbers individually from ℕ. Fail.

Because you cannot remove them -- *SETS DO NOT CHANGE*.

But the result of subtraction can be calculated. That is understood by
removing.

That is not quite the same as constructing a difference set.

It is precisely the same. And if you disagree replace "removing" by "constructing a difference set".

Regrds, WM

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Le 14/05/2024 à 16:26, Moebius a écrit :

Am 13.05.2024 um 22:21 schrieb WM:

Try to remove all natural numbers individually from ℕ.

Wie stellst Du Dir das vor? Muss man sie in Mückenhausen einzeln von
Hand removen? Das kann in der Praxis [->kein Supertask] lange dauern!

You can remove what is removable individually.

Ich hätte einen Alternativvorschlag: Man kann das mithilfe der "großen Differenz" (\\) so hinschreiben:

IN \ {1} \ {2} \ {3} \ ... = IN \\ {{1}, {2}, {3}, ...}

Ist das nicht "individuell" genug?

Individually means you can name the individuals, at least the largest
number of a removed finite initial segment.
Collectively you can use the description of a species like positive
integer.

Regards, WM

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