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

Am Mon, 20 Jan 2025 18:15:04 +0100 schrieb WM:

On 20.01.2025 15:22, FromTheRafters wrote:

WM explained :

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, ...} = { }.

All that really says is that the difference set between the set of
natural numbers and the set of natural numbers is empty.

It says that all natural numbers without ℵ₀ successors can be handled.

It does not mention nonexistent numbers.

This is different for definable natural numbers because there not all
natural numbers can be handled:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

Yes they can. The number of naturals is not a natural. Why do you
insist on using only finitely many numbers?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.