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

On 18.01.2025 14:32, Jim Burns wrote:

On 1/18/2025 3:41 AM, WM wrote:

Definitions can be seen.

>
Yes,
dark numbers however can be handled only collectively.
That distinguishes them from visible numbers.

 Definitions of visibleᵂᴹ.or.darkᵂᴹ numbers
are true claims about visibleᵂᴹ.or.darkᵂᴹ numbers
and do not distinguish them.

They do. Visible numbers can be communicated as individuals. That is not possible with all numbers. Always almost all number cannot be communicated individually. They remain.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
Nevertheless all numbers can be communicated collectively. None remains.
ℕ \ {1, 2, 3, ...} = { }

 Visible.
On paper. In chalk. Or in glowing dots.
Carved into marble. Or into clay tablets.

It is possible to communicate a visible individual. It is not possible to avoid infinitely many not communicared numbers.

⎛
⎜ Visibleᵂᴹ.or.darkᵂᴹ ordinals are
⎜ well.ordered.

Yes, but for dark numbers we cannot prove it as long as they are dark. However all dark numbers made visible turn out to be well-ordered

⎜ Visibleᵂᴹ.or.darkᵂᴹ ordinals are
⎜ linearly ordered.

Yes.

⎜
⎜ Darkᵂᴹ ordinals are linearly ordered
⎜ with respect to visibleᵂᴹ ordinals and
⎝ with respect to other darkᵂᴹ ordinals.
 Somewhere Out There, there might be
a person thinking that
"This is darkᵂᴹ"
implies
"This is something we must remain ignorant of".

Of course. All numbers larger than an infinitesimal ω/n will remain dark.

 A person thinking that would be wrong.

Try to name individually natural numbers with no rest as it can be done collectively.
Regards, WM