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

On 16.12.2024 11:23, Mikko wrote:

On 2024-12-15 11:33:15 +0000, WM said:

>
We cannot name dark numbers as individuals.

 We needn't. The axioms of natural numbers ensure that every natural number
has a successor,

The set, i.e. all numbers together, has no successor. That is a necessary condition for using all with no exception. That must happen according to
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}.

If that is not possible then there are no
natural numbers.

That is not possible for an actually infinite set. It is only possible for numbers coming into being.
Regards, WM