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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 22:42:02 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Fri, 10 Jan 2025 22:44:31 +0100 schrieb WM:
> 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.
The naturals *are* the indices of the sequence. And it is infinite.

> Therefore none can remain in the
> content. Therefore your argument is fools crap.
The limit is indeed empty.

-- 
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.