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

Am Sat, 28 Dec 2024 17:55:57 +0100 schrieb WM:

On 28.12.2024 16:44, joes wrote:

Am Sat, 28 Dec 2024 15:03:44 +0100 schrieb WM:

 

ω is the limit of the sequence 1, 2, 3, ... .

It is not a proper limit because the sequence doesn’t converge.

0 is the proper limit of the sequence of unit fractions. That can be
translated to ℕ and ω.

ω cannot be treated as a finite value.

A separate fact is that ⟦0,ω⦆ ≠ ⟦0,ω⦆∪⦃ω⦄

[0, ω-1] = [0,ω⦆ = ℕ =/= [0, ω]

Undefined. Also you have the infinity of „dark numbers” consecutive to
the naturals.

We have the infinity of dark numbers consecutive to the definable
naturals which are a (potentially in-) finite collection.

Taken the other way around, there can be only finitely many „dark
numbers”, or you don’t have all naturals. „Infinite” doesn‘t
mean „finite”.

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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.