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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)
Date: Fri, 01 Nov 2024 17:53:32 -0400
Organization: Peripheral Visions
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WM explained on 11/1/2024 :
> On 01.11.2024 19:39, FromTheRafters wrote:
>> WM formulated the question :
>
>>>> Infinite subsets don't do that for you, even if you wish really hard.
>>>
>>> They cannot evade if they are invariable.
>> 
>> Sets don't change.
>
> Therefore the elements do not depend on us and our knowledge. "If I find x, 
> then I can find x + 1" is not relevant. "For every x (that I find) there is x 
> + 1" is no relevant. All elements are there, independent of what we know or 
> do. Therefore the first and the last are also there independent of us. If 
> they weren't, their existence would depend on some circumstances and could 
> change.

Circumstances like "there is no last element"?

>>>>> It is a point on the real line, well separated from its neighbour.
>>>>
>>>> No point on the real line is separated, that is for discrete sets.
>>>
>>> All unit fractions belong to points which are separated by non-unit 
>>> fractions.
>> 
>> Okay, but you previously said the reals are separated.
>
> No, you have misunderstood. The unit fractions are separated. Of course they 
> also are real numbers.

Between any two unique unit fractions there are other fractions. They 
are not separated like the integers and the naturals. You don't need 
the reals for this subterfuge. Even you should be able to see that the 
set of denominators have no largest element to 'start' with.