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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 06:54:17 -0500
Organization: Peripheral Visions
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on 11/30/2024, WM supposed :
> 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}.
> Endsegment E(n) = {n, n+1, n+2, ...}

There it is!! Don't you see that the ellipsis means that endsegments 
are defined as infinite?
>
>> 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.

But how is that related to decreasing? What has decreased?

>> 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

Which they all are, see above.

> and therefore have an infinite intersection.

The emptyset.

>>> 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.

Of course not, since it stays at emptyset unless there is a last 
element -- which there is not since endsegments are infinite.

> 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?

Finite sets versus infinite sets. Finite ordered sets have a last 
element which can be in the intersection of all previously considered 
finite sets. Infinite ordered sets have no such last element.