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 is potential infinity, not actual infinity.

No, this is an entirely distinct concept and I don't even use those terms.

You don't know what you do.

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.
Regards, WM