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

On 09.01.2025 22:22, joes wrote:

Am Thu, 09 Jan 2025 10:30:25 +0100 schrieb WM:

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 implies the impossibility to extract all elements of contents in
order to apply them as indices.

No, you just need "extract/apply" infinitely many,

which means all natural numbers. Not even one must be missing from the
set of 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]

What does this have to do with Aleph_0?

It means that no limits are involved but that all not yet used content
of endsegments must become indices. Not all endsegments can be infinite.