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

On 01.11.2024 22:53, FromTheRafters wrote:

WM explained on 11/1/2024 :

On 01.11.2024 19:39, FromTheRafters wrote:

WM formulated the question :

>

Infinite subsets don't do that for you, even if you wish really hard.

>
They cannot evade if they are invariable.

>
Sets don't change.

>
Therefore the elements do not depend on us and our knowledge. "If I find x, then I can find x + 1" is not relevant. "For every x (that I find) there is x + 1" is no relevant. All elements are there, independent of what we know or do. Therefore the first and the last are also there independent of us. If they weren't, their existence would depend on some circumstances and could change.

 Circumstances like "there is no last element"?

That means, there is always another element. Potential infinity.

the set of denominators have no largest element to 'start' with.

If all unit fractions are existing, then a smallest unit fraction is existing. 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.
Is that too hard to understand?
Regards, WM