Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 19.01.2025 16:10, joes wrote:

Am Sun, 19 Jan 2025 11:47:50 +0100 schrieb WM:

They contain only natnumbers which have ℵ₀ successors.

There are no naturals with a finite number of successors, otherwise
you could count backwards from the end.

That is possible: ω-1, ω-2, ω-3, ...

 

If all natural numbers were applied, there would not be
successors: ℕ \ {1, 2, 3, ...} = { }.

There are no successors if only you would actually "apply" the infinitely
many naturals.

But there are always successors if you define the applied naturals individually.
Regards, WM