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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 17:08:49 +0200
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On 09.09.2024 13:49, joes wrote:
> Am Mon, 09 Sep 2024 12:27:47 +0200 schrieb WM:
>> On 08.09.2024 22:11, Richard Damon wrote:
>>> On 9/8/24 3:48 PM, WM wrote:
>>
>>>> Select any gap between one of the first ℵo unit fractions and its
>>>> neighbour. Call its size x. Then ℵo unit fractions cannot fit into the
>>>> interval (0, x), independent of the actual size.
>>> But that is changing the value of x in the middle of the problem which
>>> isn't allowed.
>>> Given that new x, we can choose a new set of Aleph_0 unit fractions
>>> below that x.
>> ℵo unit fractions are claimed to be smaller than every x > 0. If that is
>> true then I can choose as the x one of the ℵo intervals between two of
>> them.
> More precisely: every positive x has infinitely many smaller unit
> fractions (mind the quantifier order).

The quantifier order related to the problem is this: NUF(x) = ℵo means: 
There exist ℵo unit fractions smaller than any x > 0. If this is not 
true, then there are fewer. How many unit fractions are smaller than any 
x > 0. THAT is the question. None. But all are differente. Hence there 
must be a first one smaller than all other unit fractions. Note that 
real points are in question. Real points fixed on the real line.

 > A number is not an interval.

An interval has a length that can be expressed by a real number:
1/n - 1/(n+1) = x .

Then the interval (0, x) contains not all unit fractions, for instance 
not 1/n.

Regards, WM
>