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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Wed, 8 Jan 2025 10:22:15 +0100
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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