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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Sat, 30 Nov 2024 05:57:30 -0500
Organization: Peripheral Visions
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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. Almost all 
elements are considered in the new set, which means all endsegments are 
infinite.

>> all of them and each and every one of them.
>
> The set ℕ = {1, 2, 3, ..., n, n+1, ...} cannot be divided into two 
> consecutive infinite sets. As long as all endsegments are infinite, they 
> contain an infinite subset of ℕ. Therefore all indices are the finite 
> complement of ℕ.
>
>> The intersection is empty.
>
> Try to understand inclusion monotony. The sequence of endsegments decreases. 

In what manner are they decreasing? When you filter out the FISON, the 
rest, the tail, as a set, stays the same size of aleph_zero.

> 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.