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

Am 26.10.2024 um 21:35 schrieb Chris M. Thomasson:

On 10/26/2024 9:04 AM, WM wrote:

On 26.10.2024 05:21, Jim Burns wrote:

On 10/25/2024 3:15 PM, WM wrote:

>

Mainly, among other points, the claim that
all unit fractions can be defined and the claim that
a Bob can disappear in lossless exchanges.

>
The proof that all unit fractions can be defined
is to define them
as reciprocals of positive countable.to.from.0 numbers.
>
That describes all of them and only them.

>
No, you falsely assume that all natnumbers can be defined.

 Huh? Confusing to me. Humm... Are you trying to suggest that a natural number can _not_ be a natural number?

No. But most natnumbers cannot be defined. This can best be understood by the unit fractions.
Try to understand the function NUF(x) = Number of Unit Fractions between 0 and x, which starts with 0 at 0 or less and after NUF(x') = 1 cannot change to 2 without pausing for an interval consisting of uncountably many real points. The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1) > 0.
Regards, WM