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

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

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

>
Because
almost everyone has not yet realized, that
ℕ contains also the natural numbers which
cannot be counted to and which
do not leave ℵ₀ successors after being removed from ℝ.

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.
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

>
ℕ_def is a potentially infinite collection
and as such has no fxed number of elements.

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.

We use the indefinite oo in this case.

Weⁿᵒᵗᐧᵂᴹ use ℵ₀ to refer to |ℕ_def|
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
which never vanish by counting.
They can be removed only collectively
such that nothing of ℕ remains.

Without a supertask,
ℵ₀.many of anything never vanish by counting.
Fewer.than.ℵ₀ of something can vanish by counting
-- in principle, unbounded by our 13.7Gyo universe.
All the numbers which CAN be counted.to
never vanish by counting
because
each number which CAN be counted.to  is NOT
the last number which CAN be counted.to.
For NO number which can be counted.to  is
"up to that number" the same as
"all the numbers which can be counted.to".
Any possible numbers which canNOT be counted.to
are irrelevant to
that fact about numbers which CAN be counted.to.
Also,
each number which CAN be counted.to
can vanish by counting.
In summary,
there are infinitely.many finitely.preceded numbers.
Changing what ℕ refers to doesn't change that.