Re: how

On 6/3/2024 11:34 AM, Jim Burns wrote:

On 6/3/2024 7:58 AM, WM wrote:

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

>

⎜ ∀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 [positive] 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.

>
2.
Or we can assume instead that
ℕ⁺ holds all.and.only numbers countable.to by.1 from.0
ℚ⁺ holds all.and.only ratios of numbers in ℕ⁺
ℝ⁺ holds all of ℚ⁺ and all.and.only
points x between open.foresplits Fₓ and ℚ⁺\Fₓ of ℚ⁺
with no points zero distance apart
and
⅟ℕ holds all.and.only reciprocals of numbers in ℕ⁺
ℝ⁺\⅟ℕ holds all.and only the others in ℝ⁺
and
ℝ⁺ is the positive real axis.
>

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

>
Under assumption (2.)
[A] and [B] are provable for all of ⅟ℕ and ℝ⁺\⅟ℕ
>

 If A is true for dark numbers too,
then there is a positive non-unit fraction
smaller than all unit fractions.

>
Equivalent to:
If there is no positive non.unit.fraction
smaller than all unit fraction,
then A is false for darkᵂᴹ numbers too
>
And then, by (2.), no darkᵂᴹ numbers are in ℝ⁺\⅟ℕ
>

 If B is true for dark numbers too,
then there is a unit fraction
smaller than all positive non-unit fractions.

>
Equivalent to:
If there is no unit fraction
smaller than all positive non.unit.fractions,
then B is false for darkᵂᴹ numbers too
>
And then, by (2.), no darkᵂᴹ numbers are in ⅟ℕ
>

There is only one objection:

>
Versions of ℕ⁺ ℚ⁺ and ℝ⁺ which hold darkᵂᴹ numbers
are provably not the (2.) version.
>
Whatever is proved or claimed or hallucinated about
some other version is not a claim about
the (2.) version.
>

Not all subsets of unit fractions or
of non-unit fractions have two ends.

>
Pick a non.two.ended subset. 'Bye, Bob.
>

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.

>
You're too late. Bob's gone.
>
>

 The fun part is infinite congruents of continued fractions can be used to represent real numbers...? Finite congruents can only get a finite precision of a real number. :^)