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

Am 27.10.2024 um 12:38 schrieb Richard Damon:

On 10/26/24 12:04 PM, 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.

 But they have been.

Impossible. Proof: If infinity is actual, then all elements of the set of unit fractions exist. The function NUF(x) = Number of Unit Fractions between 0 and x starts with 0 at 0. After NUF(x') = 1 it cannot change to NUF(x'') =  2 without pausing for an interval consisting of uncountably many real points. The reason is this: ∀n ∈ ℕ: 1/n - 1/(n+1)  > 0. Of course x' and x'' are dark rational numbers, the smallest unit fractions.

Try to define more unit fractions than remain undefined.

 But none are.

x' and X'' are two of many.

 

Try to understand the function NUF(x) which starts with 0 at 0 and after 1 cannot change to 2 without pausing for an interval consisting of uncountably many real points.

 Bucause NUF(x) has an illogical definition, it assumes something that doesn't exist.

It assumes that all unit fractions exist. If there are no dark numbers like x' and x'', then this is clearly wrong.
Regards, WM