Re: how

Le 23/05/2024 à 21:52, Jim Burns a écrit :

On 5/23/2024 8:10 AM, WM wrote:

WM:
Between two unit fractions
there are ℵo real numbers x.

 Between any two ⅟m < ⅟n
there are more.than.any.k<ℵ₀ real.points

Even between 0 and any 1/n there are at least ℵ₀ real points.

I have shown the way: Dark numbers.

 Darkᵂᴹ numbers in ℚ and between splits of ℚ
which are between 0 and all unit fractions
do not exist, neither darklyᵂᴹ nor visiblyᵂᴹ

What is closer to zero, a unit fraction or a not unit fraction?

 We can know that they don't exist by starting with
that description and then making not.first.false
claims until we get to a contradiction.

The contradiction is ∀x ∈ (0, 1]: NUF(x) = ℵo because the unit fractions are x ∈ (0, 1].
They cannot sit at a single point x, hence the statememt is false.

 

Also true:
There is no x > 0 smaller than all unit fractions.

That implies that there is a unit fractions smaller than all other x > 0.

 

and even in accordance with
For any unit fraction there are ℵ₀ smaller real x > 0.

 Also true:
For any x > 0 there are ℵ₀ smaller unit fractions.

Impossible because the unit fractions cannot be smaller than themselves.

 

Note that
points on the real axis are fixed and
not subject to quantifier nonsense.

Regards, WM