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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: Mon, 9 Sep 2024 16:02:54 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Mon, 09 Sep 2024 17:08:49 +0200 schrieb WM:
> 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.
The wrong quantifier order is: „There is a fixed infinite set of unit
fractions, which are all less than any positive x.” The right one is:
Any positive x has AN infinite set of unit fractions less than it.
(Those sets are different; the one <0.4 is missing 1/2 compared the
set of UFs <0.75 but of course still infinite.)
How do you get to the „hence”?

>> 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.
But still infinitely many, since only finitely many are missing.

-- 
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.