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

On 1/13/2025 7:48 AM, WM wrote:

On 13.01.2025 05:51, Moebius wrote:

Es gilt dann also |ℕ|/n = |ℕ|.

>
Wrong. |ℕ| is a fixed number.

By 'fixed', you mean that #ℕ > #(ℕ\{0})
However,
ℕ is the set of all 'fixed' (finite) ordinals.
⎛ Assume ℕ is 'fixed'.
⎜
⎜ A 'fixed' ordinal ⟦0,𝔑⦆ the size of ℕ exists.
⎜ #⟦0,𝔑⦆ = #ℕ
⎜
⎜ ⟦0,𝔑+1⦆ is 'fixed', too.
⎜ ⟦0,𝔑+1⦆ ⊆ ℕ
⎜ #⟦0,𝔑+1⦆ ≤ #ℕ
⎜ #ℕ = #⟦0,𝔑⦆ < #⟦0,𝔑+1⦆ ≤ #ℕ
⎜ #ℕ < #ℕ
⎝ Contradiction.
Therefore,
ℕ is not 'fixed'.