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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: quantifier order
Date: Tue, 10 Sep 2024 21:37:05 -0400
Organization: i2pn2 (i2pn.org)
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On 9/10/24 12:24 PM, WM wrote:
> On 09.09.2024 21:55, joes wrote:
>> Am Mon, 09 Sep 2024 21:03:24 +0200 schrieb WM:
>>> On 09.09.2024 17:49, joes wrote:
> 
>>> No, that is THE question.
>> The set of unit fractions smaller than all x is empty.
> 
> Why is this so? Because all unit fractions are undercut by some x. The x 
> undercutting the last unit fractions (you said that none remains because 
> the set is empty, therefore all must be gone) do not satisfy your 
> following claim:
> 
>> However, the set of UFs smaller than one arbitrary x is infinite.
> Regards, WM
> 

Right, because any unit fraction you might want to try to nominate as 
the smallest always has at least one (actually Aleph_0) unit fractions 
smaller then it, and thus NO unit fractions are less than or equal to 
all unit fractions.

You just don't understand that mathematic of the infinite set, which 
follow rules DIFFERENT from your normal rules for finite sets.