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

On 10/28/24 4:05 PM, WM wrote:

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.

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Followed by a discontinuity.

 No. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds everywhere.

But since 1/n approaches zero, the spacing also approaches 0, and in fact faster then the numbers do, so the number of point you could put between 1/n and 0 and the spacing of 1/n and 1/(n+1) increases and the density goes to infinity.

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After NUF(x') = 1 it cannot change to NUF(x'') =  2 without pausing for an interval consisting of uncountably many real points.

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Your "Axiom of because I say so" is overworked.

 It is mathematics:>> The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0.

Which means that for even 1/n, there is a 1/(n+1) smaller than it, so no smallest 1/n.

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Non sequitur. This is just the second part of your stepwise function.

 It holds everywhere if mathematics holds everywhere.

So? The slope of NUF(x) over a range is proportial to the density of Unit Fractions in that area, since the density goes to infinity

 

It doesn't have to happen step by step as you envision it.

 As mathematics dictates it.

As FINITE mathematics dictates it, which isn't applicable, since you have an infinite set.

 You have confessed: In order to maintain actual infinity without dark numbers, you have to violate mathematics.

But, you seem to have missed that the ancients proved that we can't have "actual infinity" as something we can directly handle as finite beings.
Yes, with the finite logic of what we can directly manipulate, we can't fully instantiate the infinity, because we can't do that much work with it.
We need to move to logic that understand the infinite (which might seem a bit strange to use) and then we can mostly work with it.
You just missed the note that your Finite Logic, and Actual Infinity, just don't work together, so yes, YOU can't have actual infinity because you insist on holding on to your finite logic.

 Regards, WM