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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: Sat, 28 Dec 2024 15:03:44 +0100
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On 27.12.2024 22:00, Jim Burns wrote:
> On 12/27/2024 5:14 AM, WM wrote:

>> They are invariable numbers like ω and ω+1.
> 
> ω is
> the set of (well.ordered) ordinals k such that
> #⟦0,k⦆ ≠ #(⟦0,k⦆∪⦃k⦄)
> (such that k is finite)

That is one interpretation. My interpretation is Cantor's original one: 
ω is the limit of the sequence 1, 2, 3, ... .

> A separate fact is that
> ⟦0,ω⦆ ≠ ⟦0,ω⦆∪⦃ω⦄

[0, ω-1] = [0,ω⦆ = ℕ =/= [0, ω]

Regards, WM