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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 10 Jan 2025 20:30:19 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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.

>>> 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.

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.