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