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

On 15.12.2024 11:56, Mikko wrote:

On 2024-12-14 08:53:19 +0000, WM said:

 

Please refer to the simplest example I gave you on 2024-11-27:
The possibility of a bijection between the sets ℕ = {1, 2, 3, ...} and
D = {10n | n ∈ ℕ} is contradicted because for every interval (0, n]
the relative covering is not more than 1/10, and there are no further
numbers 10n beyond all natural numbers n.

 
It is already proven that there is such bijection. What is proven
cannot be contradicted unless you can prove that 1 = 2.

 
What is proven under false (self-contradictory) premises can be shown to
be false. Here we have a limit of 1/10 from analysis and a limit of 0
from set theory. That shows that if set theory is right, we have 1/10 =
0 ==> 1 = 0 ==> 2 = 1.

The sequence 1/10, 1/10, 1/10, ... has limit 1/10.

Irrelevant as the proof of the exitence of the bijection does not
mention that sequence.

But the disproof of the bijection does. There is no reason to forbid
that sequence.