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

On 04.01.2025 17:20, Jim Burns wrote:

On 1/4/2025 3:42 AM, WM wrote:

On 1/3/2025 3:56 PM, Jim Burns wrote:

>

All finite.ordinals removed from
the set of each and only finite.ordinals
leaves the empty set.

>
But removing
every ordinal that you can define
(and all its predecessors) from ℕ leaves
almost all ordinals in ℕ.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

>
ℕ is the set of each and only finite.ordinals.

 Yes.

|ℕ| := ℵ₀ = |ℕ\{0}| = |ℕ\{0,1}| = ... =
|ℕ\{0,1,...,n}| = ...
>
The sequence of end.segments of ℕ
grows emptier.one.by.one but
it doesn't grow smaller.one.by.one.

 It does but you cannot give the numbers because they are dark.

A precise measure must detect the loss of one element. ℵo is no precise measure but only another expression for infinitely many.

>

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

>
ℕ is the set of each and only finite.ordinals.

 Yes. But most of them cannot be named as individuals and then removed because ℵo will always remain in the set. Collectively however is works ℕ \ {1, 2, 3, ...} = { }.

>
Each finite.ordinal is not weird.
Even an absurdly.large one like Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
is not weird.

 Numbers which can be individualized are far less than 1 % of |ℕ|
 Regards, WM