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

On 08.01.2025 06:45, Jim Burns wrote:

On 1/7/2025 4:13 AM, WM wrote:

On 06.01.2025 23:43, Jim Burns wrote:

 

k ∈ ℕ  ⇒  k+1 ∈ ℕ
is true for both the darkᵂᴹ and the visibleᵂᴹ.

>
One exception exists: ω-1.

 No.
ω-1 does not exist, darkᵂᴹ or visibleᵂᴹ,
because
⦃k: k < ω ≤ k+1⦄ = ⦃⦄

If ω exists, then ω-1 exists. Then your claim is wrong.

 ⎛ Assume otherwise.
⎜ Assume ω-1 exists.
⎜ ω-1 < ω
⎜ ¬∃ᵒʳᵈψ: ω-1 < ψ < ω
⎜
⎜ However,
⎜ ω-1 ∉ ⦃⦄ = ⦃k: k < ω ≤ k+1⦄

⦃⦄ = ⦃k: k < ω ≤ k+1⦄
is a wrong presupposition.
⦃k: k < ω ≤ k+1⦄ has one element. It is dark like ω.
Regards, WM