Re: How many different unit fractions are lessorequal than all unit fractions?

On 10/4/2024 5:53 AM, WM wrote:

On 04.10.2024 10:29, joes wrote:

Am Thu, 03 Oct 2024 20:25:07 +0200 schrieb WM:

On 02.10.2024 22:07, Jim Burns wrote:

On 10/2/2024 7:12 AM, WM wrote:

On 01.10.2024 22:05, Jim Burns wrote:

For every nonzero distance d from 0
there are ℵ₀.many unit.fractions
closer than d

>
That means there is no d by what
you can distinguish ℵ₀ unit fractions?

>
You (WM) require us to guess
what you mean by 'distinguish'.

>
You distinguish two unit fractions 1/n and 1/m
if you place a point in distance d from 0
between them: 1/n < d < 1/m.

>
Which is always possible.

>
For a finite collection of unit fractions.
For almost all unit fractions it is impossible.

Yes, because
0 is the greatest.lower.bound of the set ⅟ℕᵈᵉᶜ of
decrementable.to unit.fractions
  and you know,
  for your darkᵂᴹ unit.fractions, that
0 <ᵉᵃᶜʰ ⅟ℕᵂᴹ ᵉᵃᶜʰ<ᵉᵃᶜʰ ⅟ℕᵈᵉᶜ
If ⅟ℕᵂᴹ ≠ {}
then glb.⅟ℕᵈᵉᶜ > 0
However,
glb.⅟ℕᵈᵉᶜ = 0
thus
⅟ℕᵂᴹ = {}
Slightly.fewer.than.ℵ₀.many
requires impossibilities.

There is no d between
ℵ₀.many.smaller.unit.fractions and
slightly.fewer.than.ℵ₀.many.smaller.unit.fractions.

>
That means you cannot distinguish
ℵ₀ smaller unit fractions.

>
Misconception:
there is no „slightly fewer than ℵ₀”.

If slightly fewer is not possible,
then ℵ₀ unit fractions sit at one point.

No.
Each ⅟j in ⅟ℕᵈᵉᶜ has
ℵ₀.many in ⅟ℕᵈᵉᶜ ⅟k < ⅟j
⎛ Any not.lower.bound γ of ⅟ℕᵈᵉᶜ has
⎜ ⅟j < γ and ℵ₀.many in ⅟ℕᵈᵉᶜ ⅟k < ⅟j < γ
⎜
⎜ Only a lower.bound β of ⅟ℕᵈᵉᶜ can have
⎜ fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ < β
⎜
⎜ Only a lower.bound β of ⅟ℕᵈᵉᶜ can have
⎜ fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ∪⅟ℕᵂᴹ < β
⎜ or
⎝ slightly.fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ∪⅟ℕᵂᴹ < β
However,
0 is the greatest.lower.bound of ⅟ℕᵈᵉᶜ  and
0 is one of the lower.bounds of ⅟ℕᵂᴹ
⎛ any γ > 0 has at.least.ℵ₀.many < γ
⎜ NOT slightly.fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ∪⅟ℕᵂᴹ
⎜
⎜ any β ≤ 0 has 0.many < β
⎜ NOT slightly.fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ∪⅟ℕᵂᴹ
⎜
⎜ There is NO point for which there are
⎝ slightly.fewer.than.ℵ₀.many in ⅟ℕᵈᵉᶜ∪⅟ℕᵂᴹ

If slightly fewer is not possible,
then ℵ₀ unit fractions sit at one point.

No.
Slightly fewer is not possible.
However,  ∀ᴿx>0:
⎛ uₓ(k) = ⅟⌈k+⅟x⌉
⎜ uₓ: ℕ → (0,x]: one.to.one and not.at.one.point
⎝ NUF(x) = |uₓ(ℕ)| = |ℕ| = ℵ₀
You (WM) think that that's wrong because
you (WM) think that a quantifier shift is reliable.
However, a quantifier shift is unreliable.