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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Wed, 9 Oct 2024 11:41:31 +0200
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On 08.10.2024 21:17, joes wrote:
> Am Tue, 08 Oct 2024 17:40:50 +0200 schrieb WM:
>> On 08.10.2024 15:36, joes wrote:
>>> Am Tue, 08 Oct 2024 12:40:26 +0200 schrieb WM:
>>>> On 08.10.2024 12:04, Alan Mackenzie wrote:
>>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>>
>>>>>> Hence all must be visible including the point next to zero, but they
>>>>>> are not.
>>>>> There is no point next to zero.
>>>> Points either are or are not. The points that are include one point
>>>> next to zero.
>>> But not the point inbetween?
>> If it exists then this point is next to zero.
> Ah, then the former point wasn’t the one next to zero. Same goes for this
> one. There are always infinitely many points between any two reals.
> 
>>>> The infinite sets contain what? No natural numbers? Natural numbers
>>>> dancing around, sometimes being in a set, sometimes not? An empty
>>>> intersection requires that the infinite sets have different elements.
>>> These are infinite sets: {2, 3, 4, …}, {3, 4, 5, …}, {4, 5, 6, …}.
>>> They contain all naturals larger than a given one, and nothing else.
>>> Every natural is part of a finite number of these sets (namely, its own
>>> value is that number). The set {n+1, n+2, …} does not contain n and is
>>> still infinite; there are (trivially) infinitely many further such
>>> sets. All of them differ.
>> All of them differ by a finite set of numbers (which is irrelevant) but
>> contain an infinite set of numbers in common.
> Every *finite* intersection.

As long as infinitely many numbers are captivated in endsegments, only 
finitely many indices are available, and the intersection is between 
finitely many infinite endsegments.

> Think about it this way: we are taking the limit of N\{0, 1, 2, …}.

In the limit not a single natural number remains, let alone infinitely many.
> 
>>>> Shrinking sets which remain infinite have not lost all elements.
>>> This goes for every single of these sets, but not for their infinite(!)
>>> intersection.
>> If every single set is infinite, then the intersection is infinite too.
>> These sets have lost some natural numbers but have kept infinitely many.
> 
>>> If you imagine this as potential infinity,
>> No, in potential infinity there are no endsegments.
> Uh. So the naturals don’t have successors?

They have successors but endsegments are sets and must be complete.

Regards, WM