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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: Sat, 7 Sep 2024 08:37:15 -0400
Organization: i2pn2 (i2pn.org)
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On 9/7/24 8:04 AM, WM wrote:
> On 07.09.2024 13:29, Moebius wrote:
>> Am 07.09.2024 um 11:19 schrieb Python:
>>> Le 06/09/2024 à 22:42, Crank Wolfgang Mückenheim, aka WM a écrit :
>>
>>>> [...] it is a value changing to this value + 1.
>>
>> Where exactly does it "chance"?
> 
> It, NUF(x), changes at every x = 1/n for n ∈ ℕ.
> 
> NUF(x) changes by 1 because a change by more at any x would count more 
> different unit fractions 1/n,  1/m, 1/k, ... which are identical because 
> they are the same x = 1/n = 1/m = 1/k = ... .
> 
> Regards, WM
And thus is always has a value of aleph_0 for all x > 0, since there is 
always aleph_0 unit fractions below and finite positive x value.

In doesn't "increase by one" at any value of x, because aleph_0 + 1 = 
aleph_0 by the mathematics of trans-finite numbers.

Between 0 and positive x, it just jumps, as that is an accumulation 
point for the unit fractions, which have no "smallest" value to increase 
by one at.

Thus, your verbal description of what NUF(x) should be is just incorrect 
because it is based on misconceptions.

To allow it to step the way you want, you need its domain to include a 
sub-finite set of numbers that are the reciprocals of some post-finite 
number that are abve the infinite set of Natural Numbers.

Sorry, that is just your requirements to allow NUF(x) to exist the way 
you have defined it. Since you have shown your mathematics can't even 
handle the full set of the Natural Numbers, you are not going to be able 
to handle these sub/post-finite numbers.