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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)
Date: Tue, 29 Oct 2024 06:02:27 -0400
Organization: Peripheral Visions
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WM formulated the question :
> On 28.10.2024 21:51, FromTheRafters wrote:
>> WM explained :
>>> On 28.10.2024 15:07, FromTheRafters wrote:
>>>> WM was thinking very hard :
>>>
>>>>> The function NUF(x) = Number of Unit Fractions between 0 and x starts 
>>>>> with 0 at 0.
>>>>
>>>> Followed by a discontinuity.
>>>
>>> No. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds everywhere.
>> 
>> Not for n equal to zero.
>
> I said ∀n ∈ ℕ. 0 is no natural number. NUF(0) = 0.

That is why your function is a step function with a discontinuity.

>>>>> After NUF(x') = 1 it cannot change to NUF(x'') =  2 without pausing for 
>>>>> an interval consisting of uncountably many real points.
>>>>
>>>> Your "Axiom of because I say so" is overworked.
>>>
>>> It is mathematics:>> The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0.
>>>>
>>>> Non sequitur. This is just the second part of your stepwise function.
>>>
>>> It holds everywhere if mathematics holds everywhere.
>>>
>>>> It doesn't have to happen step by step as you envision it.
>>>
>>> As mathematics dictates it.
>>>
>>> You have confessed: In order to maintain actual infinity without dark 
>>> numbers, you have to violate mathematics.
>> 
>> Bull! Just another of your non sequiturs.
>
> Try to argue mathematically in spite of spitting nonsense or be quiet.

I am, you just have trouble following along.