Re: Does the number of nines increase?

On 7/12/2024 1:04 PM, WM wrote:

Le 11/07/2024 à 02:52, Jim Burns a écrit :

On 7/10/2024 6:55 PM, Moebius wrote:

Am 11.07.2024 um 00:26 schrieb Chris M. Thomasson:

(1, 1.4, 1.41, 1.414, ...)
For each element
there is a rational that can represent it.

>
Well, A = A? Agree!

>
Just to keep things from becoming too clear...
WM has disputed A = A for darkᵂᴹ numbers A
To be precise,
he has disputed _saying_ 'A = A'

>
Dark numbers are equal to themselves.

There are no actuallyᵂᴹ.infinite sets
(different from actuallyⁿᵒᵗᐧᵂᴹ.infinite sets).
Actuallyᵂᴹ.infinite set A
has no proper subset B into which A fits
⎛ no B ≠⊂ A and f: A→B: 1.to.1
⎝ A is finiteⁿᵒᵗᐧᵂᴹ
but
actuallyᵂᴹ.infinite set A
has potentiallyᵂᴹ.infinite subset C which
has proper subset D into which C fits
⎛ yes D ≠⊂ C ⊂ A  and g: C→D: 1.to.1
⎝ C is infiniteⁿᵒᵗᐧᵂcᴹ
An actuallyᵂᴹ.infinite set is
a finiteⁿᵒᵗᐧᵂᴹ set with an infiniteⁿᵒᵗᐧᵂᴹ subset.
No actuallyᵂᴹ.infinite set exists.
because
A\C∪D is a proper subset of A  and
f: A→A\C∪D: 1.to.1
f(x) = g(x)  for x ∈ C
f(x) = x     for x ∈ A\C
No finiteⁿᵒᵗᐧᵂᴹ set with an infiniteⁿᵒᵗᐧᵂᴹ subset exists.

We cannot prove it,
because we cannot know them.
But it has to be assumed in any meaningful theory. Further
it happens to be true for
every dark number that becomes visible.

Which apparently agreesᵂᴹ with

Dark numbers are equal to themselves.

...which I guess indicates that he _agrees_ with it
and dislikes how it messes up his "proofs".

>
says a person who believes that
by exchanging two elements
one of them can be lost.

You (WM) are confused.
I have never claimed ℵ₀ = 2
ℵ₀ is the cardinality of
cardinals which, when augmented, are bigger.
If ℵ₀ is
a cardinal which, when augmented, is bigger
then
there are at least ℵ₀+1 (by assumption, more.than.ℵ₀)
cardinals which, when augmented, are bigger
...which there aren't.
ℵ₀ -- not more.than.ℵ₀ -- is the cardinality of
cardinals which, when augmented, are bigger.
Therefore,
the cardinality ℵ₀ of
cardinals which, when augmented, are bigger,
when augmented, is NOT bigger.
The set ℕ⁺ᴮᵒᵇ of
cardinals which, when augmented, are bigger plus Bob
has the same cardinality ℵ₀ as
the set ℕ of
cardinals which, when augmented, are bigger.
Bobbed ℕ⁺ᴮᵒᵇ fits in de.Bobbed ℕ
Exists f: ℕ⁺ᴮᵒᵇ→ℕ: 1.to.1
Bob ∉ f(ℕ⁺ᴮᵒᵇ)
f disappears Bob.
https://www.youtube.com/watch?v=TjAg-8qqR3g
ℵ₀ ≠ 2