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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: Tue, 10 Sep 2024 07:41:05 -0400
Organization: i2pn2 (i2pn.org)
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On 9/9/24 11:32 AM, WM wrote:
> On 09.09.2024 17:15, joes wrote:
>> Am Mon, 09 Sep 2024 16:53:32 +0200 schrieb WM:
> 
>>> You claim that ℵo unit fractions are smaller than ANY x > 0.
>> Yes. Not all the same ones of course.
> 
> My question concerns same unit fractions only. Do ℵo unit fractions 
> exist smaller than any x > 0? If not, how many same unit fractions exist 
> smaller than any x > 0? How many are smalleror equal than all unit 
> fractions?
> 
> Regards, WM
> 
> 

Whst do you mean by "same" unit fractions.

for every different x, there might be a different set of unit fractions 
below them. Those sets will have Aleph_0 unit fractions in common, but 
each are different set with possible a finite number of members different.

There is a very significant difference between asking about ANY and 
asking about ALL, especially with infinite sets.

There are Aleph_0 unit fractions below ANY x, as we can form that 
infinite set of them

There are 0 unit fractions below ALL x, as there can not be an x that is 
smaller than ALL x, and there is no smallest to be equal or less than.

It may seem strange to someone thinking in terms of finite things, but 
that is the world of infinite unbounded sets.