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From: WM <invalid@no.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Sun, 27 Oct 2024 08:38:35 +0100
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Am 26.10.2024 um 21:35 schrieb Chris M. Thomasson:
> On 10/26/2024 9:04 AM, WM wrote:
>> On 26.10.2024 05:21, Jim Burns wrote:
>>> On 10/25/2024 3:15 PM, WM wrote:
>>
>>>> Mainly, among other points, the claim that
>>>> all unit fractions can be defined and the claim that
>>>> a Bob can disappear in lossless exchanges.
>>>
>>> The proof that all unit fractions can be defined
>>> is to define them
>>> as reciprocals of positive countable.to.from.0 numbers.
>>>
>>> That describes all of them and only them.
>>
>> No, you falsely assume that all natnumbers can be defined.
> 
> Huh? Confusing to me. Humm... Are you trying to suggest that a natural 
> number can _not_ be a natural number?

No. But most natnumbers cannot be defined. This can best be understood 
by the unit fractions.

Try to understand the function NUF(x) = Number of Unit Fractions between 
0 and x, which starts with 0 at 0 or less and after NUF(x') = 1 cannot 
change to 2 without pausing for an interval consisting of uncountably 
many real points. The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0.

Regards, WM