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

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. 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.

 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.

 |ℕ_def| = ℵ₀

ℕ_def is a potentially infinite collection and as such has no fxed number of elements. We use the indefinite oo in this case.
Regards, WM