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

On 24.12.2024 19:35, Jim Burns wrote:

ℕ is the set of finite.cardinals,
all the finite.cardinals,
the visibleᵂᴹ and the darkᵂᴹ.
I gather that we don't know the name of
any darkᵂᴹ finite.cardinal δ

We do not know a finite initial segment (FISON) for any dark number. If we can construct it the number is no longer dark. But some names are known for numbers which will never become visible: ω-1, ω/2, ω/10.

Still, we know (to start with)
one fact about δ
#⟦0,δ⦆ < #(⟦0,δ⦆∪{δ})
because, being a finite.cardinal, δ ∈ ℕ
or, more.verbosely, we can write
δ  ∈  ⦃i: #⟦0,i⦆<#(⟦0,i⦆∪{i}) ⦄

That is in fact the case because dark natural numbers must have the same properties as visible numbers (if they exist at all!) because when they get visible, they behave like all visible natural numbers.

 ----
For finite sets,
emptier.by.one sets are smaller.by.one, and
fuller.by.one sets are larger.by one.
 That can serve as what.we.mean.by 'finite'

Yes.
Regards, WM