Re: how

Le 22/05/2024 à 22:48, Jim Burns a écrit :

On 5/22/2024 3:13 PM, WM wrote:

Le 22/05/2024 à 20:58, Jim Burns a écrit :

On 5/22/2024 1:57 PM, WM wrote:

Le 22/05/2024 à 17:48, Jim Burns a écrit :

 

There is no x > 0 smaller than all unit fractions.
¬∃ᴿx > 0: ∀¹ᐟᴺ ⅟k: x ≤ ⅟k

>
There is an x >= 0 smaller than all unit fractions.

 

Therefore,
you are not correct.

>
I am. It is 0.

 0 is the greatest lower bound of unit fractions.

It is an x >= 0 smaller than all unit fractions. Therefore I a correct.

Disprove this:
Between ℵo unit fractions there are at least ℵo real numbers x.
For them NUF(x) = ℵo is wrong.

 For any x > 0
there are more.than.any.k<ℵ₀ unit.fractions < x

If you are right, then there is a contradiction, since I am right with absolute certainty.
Hence we have to find a way to satisfy both statements:
WM: Between two unit fractions there are ℵo real numbers x.
JB: For any x > 0 there are ℵ₀ smaller unit fractions.
I have shown the way: Dark numbers. In accordance with:
There is no unit fraction smaller than all x > 0,
and even in accordance with
For any unit fraction there are ℵ₀ smaller real x > 0.
Note that points on the real axis are fixed and not subject to quantifier nonsense.
Regards, WM