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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? (infinitary)
Date: Sat, 2 Nov 2024 10:59:04 +0100
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On 01.11.2024 22:53, FromTheRafters wrote:
> 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"?

That means, there is always another element. Potential infinity.

> the set 
> of denominators have no largest element to 'start' with.

If all unit fractions are existing, then a smallest unit fraction is 
existing. If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions must 
be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of 
uncountably many points must be between 0 and x₀. That cannot happen at 
x₀ = 0.

Is that too hard to understand?

Regards, WM