Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

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.

 

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.
Regards, WM