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

On 5/10/2024 5:18 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. 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.

Huh? ℕ is the natural numbers, all of them. What do you mean by ℕ _also_ contains the naturals? ℕ cannot represent say, 1.12542 because its a real number... What is wrong with you!
Btw, when I say ℕ is all of the naturals does not imply that there is some magical largest natural.

>
|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