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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 23:39:21 +0100
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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

>