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

On 11/2/24 5:37 PM, WM wrote:

On 02.11.2024 21:30, Richard Damon wrote:

On 11/2/24 1:56 PM, WM wrote:

On 02.11.2024 17:02, Richard Damon wrote:

On 11/2/24 6:23 AM, WM wrote:

>

If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions must be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of uncountably many points must be between 0 and x₀. That cannot happen at x₀ = 0.

>
Right, NUF(x)is 0 at x - 0, and Aleph_0 at any x > 0, since as you said, for any positive finite x, there are Aleph_0 unit fractions below it.

>
No, neither did I say so nor is it correct. Only for x > x₀ are ℵ₀ smaller unit fractions existing.

 

So, for what finite number x > 0 is there less than that?

 Those numbers are dark.

But how are they even a number.
You are just trying to assign "black magic" to you dark numbers to blaim them for the things your logic can't handle.
They just break your logic.

>
There isn't one, as we CAN PROVE that for all x > 0, there exist and infinite number of unit fractions less that x

 For every visible number x larger, much larger than x₀ you can prove that.

Nope, I can prove it for EVERY x.
You logic assumes there is no infinity, as you can't actually have all the natural numbers. Thus your concept of "actual infinity" is just a lie.
Sorry, that *IS* what you logic is saying.

 Regards, WM