Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

Am 10.05.2024 um 20:56 schrieb Jim Burns:

On 5/10/2024 8:18 AM, WM wrote:

Le 08/05/2024 à 23:55, Jim Burns a écrit :

On 5/8/2024 3:55 PM, WM wrote:

Le 07/05/2024 à 00:11, Jim Burns a écrit :

 

All which canNOT be counted.to are not.in ℕ

>
All which canNOT be counted.to are not.in ℕ_def.

>
And all which CAN be counted.to are in ℕ_def.

>
Yes.

 Thank you.
 ℕ_def is the set of all and only
numbers which CAN be counted.to.
ℕ_def is what everyone else calls ℕ

Indeed! At least in the context of /set theory/.

ℕ contains also the natural numbers which cannot be counted to and [bla]

No, there aren't such "numbers" in ℕ. After all,
"ℕ is the set of all and only numbers which CAN be counted.to."

Weⁿᵒᵗᐧᵂᴹ use ℕ to refer to
the set of all and only
numbers which CAN be counted.to,
but we could use ℕ_def
or use ω
or use ⋃ₙ⟨⟨0…n⟩⟩
or use ♃
or use 🐎
to refer to the set of all and only
numbers which CAN be counted.to.

Indeed!
On the other hand, WM semms to "think" that somehow ℕ_def =/= ℕ, though he can't give usⁿᵒᵗᐧᵂᴹ a proper definition of ℕ_def which would allow to prove/show that (in the context of set theory).
Actually, we don't need such a definition, since the following can be proved (in the context of set theory):
AM c ℕ: ∀n ∈ M: ∃^ℵo m ∈ ℕ: m > n.
Hence if ℕ_def c ℕ (a reasonable assumption, I'd say), then we get
∀n ∈ ℕ_def: ∃^ℵo m ∈ ℕ: m > n. (*)
So WM's claim (*) concerning ℕ_def seems to be correct.
Still, "and all natural numbers which CAN be counted.to are in ℕ_def" implies ℕ c ℕ_def (as mentioned above, already).

Whichever way we refer to it,
all which canNOT be counted.to are not.in it
and all which CAN be counted.to are in it
 

|ℕ_def| = ℵ₀

Since ℕ_def = ℕ and |ℕ| = ℵ₀ (in the context of set theory).

ℕ_def is a potentially infinite collection and <bla>

@WM: There are no such entities in the context of /set theory/. Du geisteskranker Spinner.
You don't have a /theory/ of such "mathematical objects", Du hirnloser Affe. Also hör auf solche undefinierten Begriffe zu verwenden wie "potentially infinite collection".

as such has no fxed number of elements.

Maybe they are just dancing on the pin of a needle? Or we are dealing with quantum fluctuations, virtual particles? Who knows?!
Back to math:

ℕ_def  ℕⁿᵒᵗᐧᵂᴹ  ω  ⋃ₙ⟨⟨0…n⟩⟩  ♃  🐎  is
the set of all and only numbers which CAN be counted.to
and as such its elements are fixed,
because nothing exists which is
partly.countable.to and partly.not.countable.to.
Nothing is partly.in and partly.out.

At least in the context of set theory.

We use the indefinite oo in this case.

 Weⁿᵒᵗᐧᵂᴹ use ℵ₀ to refer to |ℕ_def|

Which is quite "definite".

But we could use
|ℕⁿᵒᵗᐧᵂᴹ|  |ω|  |⋃ₙ⟨⟨0…n⟩⟩|  |♃|  |🐎|
to refer to |ℕ_def|
 The claims weⁿᵒᵗᐧᵂᴹ make for
ℕ_def  ℕⁿᵒᵗᐧᵂᴹ  ω  ⋃ₙ⟨⟨0…n⟩⟩  ♃  🐎
are NOT altered by
pretending weⁿᵒᵗᐧᵂᴹ are NOT referring to
the set of all and only numbers which CAN be counted.to

But every n ∈ ℕ_def has ℵ₀ successors <bla>

Just like each and every n ∈ ℕ.