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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sun, 1 Dec 2024 15:11:41 -0800
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On 11/30/2024 3:12 AM, WM wrote:
> On 30.11.2024 11:57, FromTheRafters wrote:
>> WM explained :
>>> On 29.11.2024 22:50, FromTheRafters wrote:
>>>> WM wrote on 11/29/2024 :
>>>
>>>>> The size of the intersection remains infinite as long as all 
>>>>> endsegments remain infinite (= as long as only infinite endsegments 
>>>>> are considered).
>>>>
>>>> Endsegments are defined as infinite,
>>>
>>> Endsegments are defined as endsegments. They have been defined by 
>>> myself many years ago.
>>
>> As what is left after not considering a finite initial segment in your 
>> new set and considering only the tail of the sequence.
> 
> Not quite but roughly. The precise definitions are:
> Finite initial segment F(n) = {1, 2, 3, ..., n}.

Finite? Huh? The natural numbers don't stop at n! WTF!!!!  Lay off the 
drugs.


> Endsegment E(n) = {n, n+1, n+2, ...}





> 
>> Almost all elements are considered in the new set, which means all 
>> endsegments are infinite.
> 
> Every n that can be chosen has infinitely many successors. Every n that 
> can be chosen therefore belongs to a collection that is finite but 
> variable.
> 
>>> Try to understand inclusion monotony. The sequence of endsegments 
>>> decreases. 
>>
>> In what manner are they decreasing?
> 
> They are losing elements, one after the other:
> ∀k ∈ ℕ : E(k+1) = E(k) \ {k}
> But each endsegment has only one element less than its predecessor.
> 
>> When you filter out the FISON, the rest, the tail, as a set, stays the 
>> same size of aleph_zero.
> 
> For all endsegments which are infinite and therefore have an infinite 
> intersection.
>>
>>> As long as it has not decreased below ℵo elements, the intersection 
>>> has not decreased below ℵo elements.
>>
>> It doesn't decrease in size at all.
> 
> Then also the size of the intersection does not decrease.
> Look: when endsegments can lose all elements without becoming empty, 
> then also their intersection can lose all elements without becoming 
> empty. What would make a difference?
> 
> Regards, WM
>