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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, 19 Dec 2024 15:50:28 +0100
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On 18.12.2024 21:21, Jim Burns wrote:
> On 12/18/2024 2:14 PM, WM wrote:
>> On 18.12.2024 19:22, Jim Burns wrote:
>>> On 12/18/2024 7:40 AM, WM wrote:
>>>> On 18.12.2024 02:16, Jim Burns wrote:
> 
>>>>> The infinite end.segments of finite.cardinals
>>>>> do not include any finite end.segments and
>>>>> they have an empty intersection.
>>>>
>>>> Explicitly wrong.
>>>> As long as
>>>> only infinite endsegments are concerned
>>>> their intersection is infinite.
>>>> There are no limits involved.
>>>
>>> Then it would be great if you (WM)
>>> stopped calling things 'limit sets'
>>
>> The sets described here are not limit sets.
> 
> Q. What is a limit set?
> 
>> f(k) = ∩{E(1), E(2), ..., E(k)}
>> with E(1) = ℕ
>> ∀k ∈ ℕ :
>> ∩{E(1), E(2), ..., E(k+1)} =
>> ∩{E(1), E(2), ..., E(k)} \ {k},
>> ∩{E(1), E(2), ...} is empty.
>> Other sets are not in the argument.
> 
> ∀k ∈ ℕ :
> k ∉ E(k+1) ⊇ ⋂{E(1),E(2),...} ∌ k
> 
> ⋂{E(1),E(2),...} is empty.

But E(1) is full. The only way to get rid of content is to proceed by
∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k},
i.e. to lose one number per term of the function
f(k) = ∩{E(1), E(2), ..., E(k)}.

The empty term has lost all natnumbers one by one.

Regards, WM