Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/27/2024 12:13 PM, WM wrote:

On 27.11.2024 13:32, Richard Damon wrote:

[...]

>
Yes, the same number of elements,
but not the same number of natural numbers.

A finite cardinal can change by 1
⎛ Finite cardinal Ψ requires the existence of
⎝ finite cardinals Ψ+1 Ψ-1 which aren't Ψ
ℕᶠⁱⁿ is the set of finite cardinals.
For each finite cardinal Ψ in ℕᶠⁱⁿ
⎛ ℕᶠⁱⁿ has subset |⟦0,Ψ-1⟧| = Ψ
⎜ ℕᶠⁱⁿ has subset |⟦0,Ψ⟧| = Ψ+1 > Ψ
⎜ |ℕᶠⁱⁿ| ≥ |⟦0,Ψ⟧| = Ψ+1 > Ψ
⎝ |ℕᶠⁱⁿ| ≠ Ψ
|ℕᶠⁱⁿ| isn't any finite cardinal.
Finite cardinals can change by 1
The cardinal |ℕᶠⁱⁿ| cannot change by 1
Yes,
the sets can change membership by 1
(for a different set)
ℕᶠⁱⁿ ≠ Eᶠⁱⁿ(1) ≠ Eᶠⁱⁿ(2) ≠ Eᶠⁱⁿ(3) ≠ ...
However,
the cardinalities of those sets cannot change by 1
|ℕᶠⁱⁿ| = |Eᶠⁱⁿ(1)| = |Eᶠⁱⁿ(2)| = |Eᶠⁱⁿ(3)| = ...
ℕᶠⁱⁿ, Eᶠⁱⁿ(1), Eᶠⁱⁿ(2), Eᶠⁱⁿ(3), ...
are infinite sets,
NOT darkly.large.but.countable.to sets.