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

On 1/20/25 7:29 AM, WM wrote:

On 19.01.2025 13:32, Richard Damon wrote:
 

Note, even in actual infinity, every Natural Number has Aleph_0 successors

 Then not all could be subtracted from ℕ. But that is possible  ℕ \ {1, 2, 3, ...} = { }.
 Regards, WM
 

So? That set is off ALL, not ANY.
Your logic just doesn't have the words to handle the infinite sets.
Yes, EVERY Natural Number has Aleph_0 successors.
So, it is true for ANY Natural Number, that it has Aleph_0 successors.
Gather ALL of them, into an infinite set, and that infinite operation changes things, and now we have a infinite set that spans the infinite domain of the Natural Numbers.
If you logic can't handle the diffence between ANY and ALL, it can't handle infinite sets.
Note, there isn't a "Natural Nubmber" that ... represents, that represents an INFINITE sequence of them, so your logic doesn't apply.