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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, infinite-middle)
Date: Sun, 1 Dec 2024 15:17:18 -0800
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On 11/30/2024 10:27 AM, Ross Finlayson wrote:
> On 11/30/2024 03:54 AM, FromTheRafters wrote:
>> 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.
> 
> What about the "infinite-middle" models?

{ 1, ..., 1.5, ... 2, ... 2.5, ..., 3, ... }

There are infinite reals in each (...)

?


> 
> This is simply about a symmetric rather than a-symmetric
> outset of integers, for example.
> 
> As "sets", with their "ordering", and by that I mean sets
> with an ordering, there's first and last, alpha and omega
> as it were.
> 
>