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

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⦄ = ⦃⦄
⎛ Assume otherwise.
⎜ Assume ω-1 exists.
⎜ ω-1 < ω
⎜ ¬∃ᵒʳᵈψ: ω-1 < ψ < ω
⎜
⎜ However,
⎜ ω-1 ∉ ⦃⦄ = ⦃k: k < ω ≤ k+1⦄
⎜ ¬(w-1 < w ≤ (w-1)+1)
⎜ w-1 < (w-1)+1 < w
⎜ ∃ᵒʳᵈψ: ω-1 < ψ < ω
⎝ Contradiction.
Therefore,
ω-1 does not exist.

(0, ω)*2 = {2, 4, 6, ..., ω, ω+2, ω+4, ...}.

No.
⦃k: j,k < ω ≤ j⋅k⦄ = ⦃⦄
⦃k: k < ω ≤ 2⋅k⦄ = ⦃⦄
2⋅ᵉᵃᶜʰ⦅0, ω⦆ ∩ ⦃ω,ω+2,ω+4,...⦄ = ⦃⦄
----
⎛ Define k < ω  ⇔  #⟦0,k⦆ < #⟦0,k+1⦆
⎜
⎜ #A < #B  ⇒  #(A∪{a}) < #(B∪{b})
⎜
⎜ #⟦0,k⦆ < #⟦0,k+1⦆  ⇒  #⟦0,k+1⦆ < #⟦0,k+2⦆
⎜
⎜ k < ω  ⇒  k+1 < ω
⎜
⎜ ¬(k < ω ≤ k+1)
⎜
⎝ ⦃k: k < ω ≤ k+1⦄ = ⦃⦄
⎛ Define j+(k+1) = (j+k)+1
⎜
⎜ ⦃k: j,k,k+1,j+k < ω ≤ (j+k)+1⦄  ⊆  ⦃k: k < ω ≤ k+1⦄
⎜
⎜ ⦃k: j,k+1,j+k < ω ≤ j+(k+1)⦄  ⊆  ⦃⦄
⎜
⎜ ⦃k: j,k+1,j+k < ω ≤ j+(k+1)⦄  =  ⦃⦄
⎜
⎜⎛ Assume ⦃k: j,k+1 < ω ≤ j+(k+1)⦄  ≠  ⦃⦄
⎜⎜
⎜⎜ k₀ = min.⦃k: j,k+1 < ω ≤ j+(k+1)⦄
⎜⎜
⎜⎜ j,k₀+1,j+k₀ < ω
⎜⎜ j,k₀+1 < ω ≤ j+(k₀+1)
⎜⎜ k₀ ∈ ⦃k: j,k+1,j+k < ω ≤ j+(k+1)⦄
⎜⎜
⎜⎜ However,
⎜⎜ ⦃k: j,k+1,j+k < ω ≤ j+(k+1)⦄  =  ⦃⦄
⎜⎝ Contradiction.
⎜
⎜ ⦃k: j,k+1 < ω ≤ j+(k+1)⦄  =  ⦃⦄
⎝ ⦃k: j,k < ω ≤ j+k⦄  =  ⦃⦄
⎛ Define j⋅(k+1) = (j⋅k)+j
⎜
⎜ ⦃k: j,k,k+1,j⋅k < ω ≤ (j⋅k)+j⦄  ⊆  ⦃k: j,k < ω ≤ j+k⦄
⎜
⎜ ⦃k: j,k+1,j⋅k < ω ≤ j⋅(k+1)⦄  =  ⦃⦄
⎜
⎜⎛ Assume ⦃k: j,k+1 < ω ≤ j⋅(k+1)⦄  ≠  ⦃⦄
⎜⎜
⎜⎜ k₀ = min.⦃k: j,k+1 < ω ≤ j⋅(k+1)⦄
⎜⎜
⎜⎜ k₀ ∈ ⦃k: j,k+1,j⋅k < ω ≤ j⋅(k+1)⦄
⎜⎜
⎜⎜ However,
⎜⎜ ⦃k: j,k+1,j⋅k < ω ≤ j⋅(k+1)⦄  =  ⦃⦄
⎜⎝ Contradiction.
⎜
⎜ ⦃k: j,k+1 < ω ≤ j⋅(k+1)⦄  =  ⦃⦄
⎝ ⦃k: j,k < ω ≤ j⋅k⦄  =  ⦃⦄