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

On 10.01.2025 21:30, joes wrote:

Am Thu, 09 Jan 2025 23:39:21 +0100 schrieb WM:

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.

In particular it means there is no largest one.

Relevant is only that none remains outside of the set of indices. It would make the set finite.

 

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.

Yes they can, because there are an infinity of them.

That is wrong. Infinitely many of them can only exist when no natural natural number is missing an an index. Therefore none can remain in the content. Therefore your argument is fools crap.
Regards, WM