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

Am Sat, 28 Dec 2024 15:03:44 +0100 schrieb WM:

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, ... .

It is not a proper limit because the sequence doesn’t converge.

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

[0, ω-1] = [0,ω⦆ = ℕ =/= [0, ω]

Undefined. Also you have the infinity of „dark numbers” consecutive
to the naturals.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.