Re: Replacement of Cardinality

On 8/24/2024 10:15 AM, WM wrote:

Le 23/08/2024 à 20:42, Jim Burns a écrit :

On 8/23/2024 10:32 AM, WM wrote:

Le 22/08/2024 à 19:24, joes a écrit :

Am Thu, 22 Aug 2024 11:07:42 +0000 schrieb WM:

Le 21/08/2024 à 20:14, joes a écrit :

The unit fractions get denser toward 0.
They are all positive.

>
They all have gaps.
This implies a last one if all are existing.

>
How so?

>
Every point is either in the empty domain
or in the populated domain.

>

One point is the last one in the populated domain.

>
Only in some cases.

 In cawsews with distances.

⎛ Open sets exist.
⎝ An open set doesn't hold any of its boundary.

Split ℚ⁺ the positive rationals into {p²≤2}ꟴ, {2≤p²}ꟴ
No rational √2 exists.

>
But the last rational approximation exists.
It is dark.

⎛ A rational.approximation is best
⎜ iff
⎝ no smaller.denominated approximation is better.
An endless sequence (holding no last) exists
of best rational.approximations to √​̅2 with
numerators and denominators from ℕᵈᵉᶠ
1/1, 3/2, 7/5, 17/12, 41/29, 99/70, ...
Endless:
A last (closest) best ℕᵈᵉᶠ.approximation
does not exist.
⎛ The rational.sequence is generated from
⎜ the endless ℕᵈᵉᶠ.sequence
⎜ 0, 1, 2, 5, 12, 29, 70, ...
⎜ which has the recursion relation qₙ = 2⋅qₙ₋₁+qₙ₋₂
⎜
⎜ The '2' comes from
⎝ √​̅2-1 = ⅟(2+(√​̅2-1)) = ⅟(2+⅟(2+(√​̅2-1))) = ...
The approximations 1+qₙ₋₁/qₙ alternate
above and below √​̅2
For each non.√​̅2 point,
a better 1+qₙ₋₁/qₙ exists: qₙ₋₁,qₙ ∈ ℕᵈᵉᶠ
⟹
A last (closest) rational approximation does not exist.

But the last rational approximation exists.
It is dark.

No.
A last (closest) rational approximation does not exist.
----
For each non.√​̅2 point,
a better 1+qₙ₋₁/qₙ exists: qₙ₋₁,qₙ ∈ ℕᵈᵉᶠ
The core argument is the same as for the claim that
no x > 0 is a lower.bound of ⅟ℕᵈᵉᶠ visibleᵂᴹ.unit.fractions
but it uses more algebra.
⎛ Assume otherwise.
⎜ Assume positive β = glb of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
⎜
⎜ γ is a little larger than β
⎜ γ isn't lower.bound of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
⎜ 1+qₖ₋₁/qₖ is closer than γ to √​̅2
⎜
⎜ 1+qₖ/qₖ₊₁ is closer than 1+qₖ₋₁/qₖ
⎜ 1+qₖ₊₁/qₖ₊₂ is closer than 1+qₖ/qₖ₊₁
⎜ 1+qₖ₊₁/qₖ₊₂ is closer than α such that
⎜ α is a little smaller than β
⎜
⎜ α is smaller than β
⎜ α is lower.bound of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
⎜ 1+qₖ₊₁/qₖ₊₂ is closer than α
⎜ α is not.lower.bound of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
⎝ Contradiction.
Therefore,
0 = glb of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
If non.√​̅2 point δ existed such that
no better 1+qₙ₋₁/qₙ exists: qₙ₋₁,qₙ ∈ ℕᵈᵉᶠ
|δ-√​̅2| would be
positive lower.bound of allᵈᵉᶠ|1+qₙ₋₁/qₙ-√​̅2|
Since there isn't positive lower.bound |δ-√​̅2|
there isn't δ better than all 1+qₙ₋₁/qₙ
And
a last (closest) rational approximation does not exist.

Even so,
each two rationals are separated by a distance > 0.

>
Of course.
But the necklace analogon is easier to comprehend.

Many of us are more familiar with necklaces which
have some minimum size of bead (or of whatever).
For rationals,
there is a greatest lower bound of distances
between different rationals,
but,
for a minimum distance to exist,
the greatest lower bound needs to be a distance.
(There aren't two greatest lower bounds.)
The greatest lower bound of distances is 0,
and 0 is not a distance between two rationals.
The greatest lower bound is not the minimum.
Nothing else is the minimum.
The minimum does not exist...
...unlike all the necklaces I'm familiar with.