Re: how

On 5/23/2024 8:10 AM, WM wrote:

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

[...]

>
Hence
we have to find a way to satisfy both statements:

JB:
For any x > 0
there are ℵ₀ smaller unit fractions.

For any x > 0
there are more.than.any.k<ℵ₀ unit.fractions < x
Among them are
more.than.k  ⅟⌊(1+⅟x)⌋  ...  ⅟⌊(k+1+⅟x)⌋

WM:
Between two unit fractions
there are ℵo real numbers x.

Between two unit fractions ⅟m ⅟n
there are more.than.any.k<ℵ₀ real numbers.
Among them are
more.than.k  1⋅dₘₙₖ+⅟m  ...  (k+1)⋅dₘₙₖ+⅟m
dₘₙₖ = (⅟n-⅟m)/(k+2)
∀ᴿ x>0: ∀ᶜᵃʳᵈk<ℵ₀:  k<|⅟ℕ∩(0,x)|   [1]
∀¹ᐟᴺ ⅟m ⅟n: ∀ᶜᵃʳᵈk<ℵ₀:  k<|(⅟m,⅟n)|   [2]
∀ᴿ x>0:  ℵ₀≤|⅟ℕ∩(0,x)|
∀¹ᐟᴺ ⅟m ⅟n:  ℵ₀≤|(⅟m,⅟n)|
| Assume otherwise.
| Assume
| |⅟ℕ∩(0,x₁)| = k₁ < ℵ₀
| |(⅟m₂,⅟n₂)| = k₂ < ℵ₀
|
| From [1], for k1 = |⅟ℕ∩(0,x₁)|
| |⅟ℕ∩(0,x₁)| < |⅟ℕ∩(0,x₁)|
| From [2], for k₂ = |(⅟m₂,⅟n₂)|
| |(⅟m₂,⅟n₂)| < |(⅟m₂,⅟n₂)|
|
| However,
| ¬( |⅟ℕ∩(0,x₁)| < |⅟ℕ∩(0,x₁)| )
| ¬( |(⅟m₂,⅟n₂)| < |(⅟m₂,⅟n₂)| )
| Contradictions.
Therefore,
∀ᴿ x>0:  ℵ₀≤|⅟ℕ∩(0,x)|
∀¹ᐟᴺ ⅟m ⅟n:  ℵ₀≤|(⅟m,⅟n)|

I have shown the way: Dark numbers.
In accordance with:
There is no unit fraction smaller than all x > 0,

Also true:
There is no x > 0 smaller than all unit fractions.
¬∃ᴿx>0: ∀¹ᐟᴺ ⅟m: x≤⅟m
| Assume otherwise.
| Assume x₃>0: ∀¹ᐟᴺ ⅟m: x₃≤⅟m
|
| ¬∃¹ᐟᴺ ⅟m: ⅟m<x₃
| b₃ is the least upper bound of
| {x∈ℝ: ¬∃¹ᐟᴺ ⅟m: ⅟m<x}
|
| 0 < x₃ ≤ b₃
| 0 < ½⋅b₃ < b₃ < 2⋅b₃
| ¬∃¹ᐟᴺ ⅟m: ⅟m<½⋅b₃
| ∃¹ᐟᴺ ⅟m: ⅟m<2⋅b₃
|
| However,
| exists ⅟m₃<2⋅b₃
| (⅟m₃)/4<(2⋅b₃)/4
| exists ⅟(4⋅m₃)<½⋅b₃
| ∃¹ᐟᴺ ⅟m: ⅟m<½⋅b₃
| Contradiction.
Therefore,
There is no x > 0 smaller than all unit fractions.

and even in accordance with
For any unit fraction
there are ℵ₀ smaller real x > 0.

There is no real x > 0 for which
there are fewer.than.ℵ₀ smaller unit fractions.
| Assume otherwise.
| Assume |⅟ℕ∩(0,x₄)| = k₄ < ℵ₀
|
| k₄ is finite.
| Exists ⅟G smallest in ⅟ℕ∩(0,x₄)
| ⅟(2⋅G) > 0 is smaller than all unit fractions.
|
| However, above, we see that
| there is no x > 0 smaller than all unit fractions.
| Contradiction.
Therefore,
There is no real x > 0 for which
there are fewer.than.ℵ₀ smaller unit fractions.

Note that points on the real axis are fixed and
not subject to quantifier nonsense.

How variables and quantifiers work and
how you (WM) think variables and quantifiers work
are different.