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

On 13.01.2025 03:54, Richard Damon wrote:

On 1/12/25 5:58 AM, WM wrote:

On 11.01.2025 14:34, joes wrote:
>

1, 2, 3, 4, 5, ..., ω becomes 2, 4, 6, 8, 10, ..., ω, ω+2, ω+4, ..., ω2.

No. There is no x e N such that 2*x >= omega. You have listed two
consecutive infinities on the right.

>
There is a basic law: When a sequence of regular distances is multiplied by 2, then a sequence of regular distances results.

 Which isn't applicable, since their isn't such a "regular distance"

Between all natural numbers, there is a regular distance. When doubled, the new ones do not fit between the old ones and ω because nothing fits between ℕ and ω.

your starting premise is just

to double all numbers which fit between 1 and ω.

The interval (0, ω)*2 becomes (0, ω*2) with ω in the middle. Below ω the newly created even numbers cannot be inserted, because more than all even natural numbers do not exist in actual infinity.
>
Every contrary opinion is based on potential infinity.
>

 Nope, your "opinion" is just based on your

correct understanding that all between 1 and ω is available for doubling and nothing can be inserted.
Regards, WM