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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Mon, 9 Sep 2024 07:47:59 -0400
Organization: i2pn2 (i2pn.org)
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On 9/9/24 6:31 AM, WM wrote:
> 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.

But there is always room at the bottom, where the gaps keep getting smaller

>>
>> 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
> 
> 

Nope, I guess you don't understand what you claim means.

How can a claim a set of values less then a SPECIFIC NUMBER if I don't 
have the number?

Sorry, you are just proving your stupidity. YOuor logic is like proving 
you can't add two numbers together to get the sum, which you try to 
prove by asking for the sum, and then provide the two numbers that don't 
add up to it.