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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, 28 Sep 2024 07:24:03 -0400
Organization: i2pn2 (i2pn.org)
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On 9/27/24 2:54 PM, WM wrote:
> On 25.09.2024 20:40, Jim Burns wrote:
>> On 9/25/2024 11:51 AM, WM wrote:
> 
>>> That means NUF increases by 1
>>> at every point occupied by a unit fraction.
>>
>> There are numbers (cardinalities) which increase by 1
>> and other numbers (cardinalities), which
>> don't increase by 1.
> 
> No. Every countable set is countable, i.e., it increases one by one.

Its indexes increase one by one.

And for a countably INFINITE set, like the Natural Numbers, there is 
only one end you can count from.

Your problem is you try to make the Naturals just "finite countable", 
which has the sort of properties you can handle, when they are actually 
infinitely countable.


> 
>> For each positive point x
>> for each number (cardinality) k which can increase by 1
>> there are more.than.k unit.fractions between 0 and x
> 
> That is a misinterpretation of the law valid for small numbers.

No, it applies for *ALL* numbers.

>>
>> For each positive point x
>> the number (cardinality) of unit.fractions between 0 and x
>> is not
>> any number (cardinality) which increases by 1
>> Instead, it is
>> a number (cardinality) which doesn't increase by 1.
> 
> For every x NUF increases by not more than 1.

No, for every finite x, NUF doesn't increase at all, since for ALL 
finite values x > 0, it has already reached an infinte value of Aleph_0, 
and that doesn't increase when incremented.

> 
>> Each positive point is undercut by
>> some finite.unit.fraction.
> 
> Repetition is apparently what you (JB) think mathematics is!

Nope,  that is YOUR failing, you keep repeating your own lies that come 
out of your ignorance.

> 
> Prove that ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 is wrong or agree.

It it true, and proves that there is no smallest unit fraction as it 
says for EVERY 1/n there is a 1/(n+1) < 1/n, and since for all n, there 
IS a n+1 (as part of the DEFINITION of the Natural Numbers) so you claim 
is just disproven, and you are shown to be a stupid liar.

Of course, your responce will be that there isn't always an n+1, but 
that just proves you don't understand what the Natural Nubmers are, and 
never did, and thus your whole premise is based on ignorant lies.

> 
> Regards, WM