Re: how

Le 03/06/2024 à 10:57, Jim Burns a écrit :

On 6/1/2024 11:15 AM, WM wrote:

Le 31/05/2024 à 21:15, Jim Burns a écrit :

On 5/31/2024 1:15 PM, WM wrote:

 

Ask colleagues
(without pointing to our discussion)
whether they agree that
in the course of exchanging elements,
infinitely many elements can disappear.
Ask further whether
in the accumulation point of the sequence (1/n)
infinitely many unit fractions
can populate one and the same point.

>
Even better:
Ask whether
∀ᴿ⁺y ∃ᴿ⁺x≠y: x<y   implies
∃ᴿ⁺x ∀ᴿ⁺y≠x: x<y

>
No.

 "No", what?

Not better.

⎜ ∀ᴿ⁺y ∃ᴿ⁺x≠y: x<y   implies
⎝ ∃ᴿ⁺x ∀ᴿ⁺y≠xv : x<y

No this is not implied but independently proven in
Evidence for Dark Numbers, prepublished chapter 4.2:
We assume that all points on the real axis are fixed and can be subdivided into two sets, namely the set of unit fractions and the set of positive non-unit fractions. For visible numbers we have two statements both of which are true:
[A] There is no unit fraction smaller than all positive non-unit fractions.
[B] There is no positive non-unit fraction smaller than all unit fractions
 If A is true for dark numbers too, then there is a positive non-unit fraction smaller than all unit fractions.
 If B is true for dark numbers too, then there is a unit fraction smaller than all positive non-unit fractions.
There is only one objection: Not all subsets of unit fractions or of non-unit fractions have two ends. But this is dismissed by the fact that the positive real axis and all point sets in it have an end at or before zero.
Regards, WM