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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, 9 Jan 2025 10:30:25 +0100
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On 09.01.2025 00:42, joes wrote:
> Am Wed, 08 Jan 2025 15:35:44 +0100 schrieb WM:

>> A set like ℕ has a fixed number of elements. If ω-1 does not exist, what
>> is the fixed border of existence?
> It has an infinite number of elements, and that number happens to be
> invariant under finite subtraction/addition.

That is potential infinity, not actual infinity.
Invariability under finite subtraction implies the impossibility to 
empty the endsegments. That implies the impossibility to extract all 
elements of contents in order to apply them as indices. That destroys 
Cantor's approach. His sequences do not exist:
"thus we get the epitome (ω) of all real algebraic numbers [...] and 
with respect to this order we can talk about the nth algebraic number 
where not a single one of this epitome (ω) has been forgotten." [E. 
Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und 
philosophischen Inhalts", Springer, Berlin (1932) p. 116]

Regards, WM