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

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