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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 9 Oct 2024 17:47:31 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Wed, 09 Oct 2024 18:56:29 +0200 schrieb WM:
> Am 09.10.2024 um 18:12 schrieb Alan Mackenzie:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>>>> Dark numbers don't exist, or at least they're not natural numbers. 
>>>> There is no number in each and every end segment of N. 
>>> True. But those endsegments which have lost only finitely many numbers
>>> and yet contain infinitely many, have an infinite intersection.
>> End segments don't "lose" anything.  They are what they are, namely
>> well defined sets.  Note that your "True" in your last paragraph,
>> agrees that the intersection of all end segments is empty, which you
>> immediately contradict by asserting it is not empty.
> You do not understand the least! The intersection of all endsegments is
> empty. The intersection of infinite endsegments is infinite.
What does this --------------^ specify exactly that distinguishes it from
the preceding sentence? Especially since both all segments are infinite,
and there are infinitely many of them.

>>>>>>> Note: The shrinking endsegments cannot acquire new numbers.
>>>>>> An end segment is what it is.  It doesn't change.
>>>>> But the terms of the sequence do. Here is a simple finite example:
>>>>> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
>>>>> {2, 3, 4, 5, 6, 7, 8, 9, 10}
>>>>> {3, 4, 5, 6, 7, 8, 9, 10}
>>>>> {4, 5, 6, 7, 8, 9, 10}
>>>>> {5, 6, 7, 8, 9, 10}
>>>>> {6, 7, 8, 9, 10}
>>>>> {7, 8, 9, 10}
>>>>> {8, 9, 10}
>>>>> {9, 10}
>>>>> {10}
>>>>> { } .
>>>>> Theorem: Every set that contains at least 3 numbers (call it TN-set)
>>>>> holds these numbers in common with all TN-sets.
>>>>> Now complete all sets by the natural numbers > 10 and complete the
>>>>> sequence.
>> Then you get different sets, which weren't the ones you were trying to
>> reason about.
> The completion of the above sets does not change the principle:
> Non-empty inclusion-monotonic sets like infinite endsegments have a
> non-empty intersection. All endsegments have an empty intersection.
You should really be more careful with your phrasing. Intersection with 
what?

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.