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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Sun, 6 Oct 2024 11:41:18 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
Message-ID: <864fcf57e8b06354bfb8c125be76f6e9b1f4e5fb@i2pn2.org>
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Am Sat, 05 Oct 2024 20:54:21 +0200 schrieb WM:
> On 05.10.2024 15:02, joes wrote:
>> Am Sat, 05 Oct 2024 11:32:44 +0200 schrieb WM:
>>> On 05.10.2024 10:08, joes wrote:
>>>> Am Fri, 04 Oct 2024 22:09:41 +0200 schrieb WM:
>>>>> But it is no quantifier shift but simplest logic:
>>>>> If ℵ₀ unit fractions do not sit at one point (and are not dark) then
>>>>> they can be subdivided into smaller parts.
>>>> Which they can.
>>> Do it. Fail because slightly fewer is not possible.
>> What do you mean? All the unit fractions have, as you say,
>> finite distances, which means there are numbers inbetween.
> But these numbers and these unit fractions cannot be found. It is
> impossible to define a unit fraction having less than ℵo smaller unit
> fractions. It is impossible to define a unit fraction being closer to
> zero although it is obvious that there are points closer to zero than
> ℵo*2^ℵo points.
Yes, the can be found, for example by the arithmetic mean. There simply
are no unit fractions with a finite number of lesser UFs, otherwise those
that are greater couldn’t have an infinite number less than them. You 
cannot start from zero, count a finite number of UFs, and reach an
infinity in ever greater steps. It is always possible to increase the
denominator to get a UF closer to 0. Every point has a finite distance
from zero, i.e. an infinite number of points inbetween.

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.