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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Thu, 26 Dec 2024 18:34:02 +0100
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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