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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: Mon, 9 Sep 2024 12:31:55 +0200
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On 08.09.2024 22:11, Richard Damon wrote:
> On 9/8/24 3:52 PM, WM wrote:
>> On 07.09.2024 16:51, Richard Damon wrote:
>>> On 9/7/24 9:45 AM, WM wrote:
>>
>>>> Because ℵo unit fractions have ℵo gaps. We can use one of the gaps 
>>>> as x  > 0.
>>>>
>>> But you started with that x, so it was fixed to begin with.
>>
>> No, it is the size of one of the gaps between two adjacent unit 
>> fractions, ℵo of which are claimed to fit into (0, x).
> 
> So it is the size of *ONE* of the gaps, you need to choose which one, as 
> all the gaps are different sizes.

And all gaps are occupied by the unit fractions. Hence every gap is too 
small.
> 
> Remember, just as there is no smallest unit fraction, there is no 
> smallest gap, so you can't specify choosing the smallest.
> 
> The fact that the choice is for ANY, it requires you to actually CHOOSE 
> one, and then we can find the Aleph_0 smaller.

No. First you claim the unit fractions with their gaps. Then I choose 
one of them, irrelevant which one. Each one is smaller than all.

Regards, WM