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

On 9/7/24 7:02 AM, WM wrote:

On 07.09.2024 01:51, Jim Burns wrote:

On 9/6/2024 4:52 PM, WM wrote:

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Between 0 and your defined x or epsilon,
not between 0 and every possible x.

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If x > 0  then there is ⅟⌊1+⅟x⌋  between 0 and x

 Then identical unit fractions differ.
They are identical because NUF(x) counts them at the same x, but they differ because NUF(x) counts more than 1 at this x.

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A quantifier shift is not reliable.

 That is no quantifier shift but simplest logic.

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Between 0 and x
there are more.than.any.k<ℵ₀ unit fractions.

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Between 0 and every epsilon you can define.

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Between 0 and any epsilon satisfying my description.
I haven't made a claim for other epsilons.

 Fine. There I agree. Every of your epsilons has ℵo smaller unit fractions in (0, eps). This proves dark numbers.
 Regards, WM

No, not the way you try to call them. It proves your concepts are just broken.
Yes, we might be able to define a set of "dark numbers" as being a type of sub-finite and post-finite number, the sub-finite numbers being a form of infinitesimal smaller than any finite number, and the post-finite infinite numbers that are greater than all of the infinite set o finite numbers, but you seem to want to try to define that possibility away, and since your mathematical ability can't even handle the unbounded finite numbers, it has no hope for these sub/post-finite values.