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From: WM <wolfgang.mueckenheim@tha.de>
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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