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

On 9/19/2024 1:29 PM, Jim Burns wrote:

On 9/19/2024 12:29 AM, Ross Finlayson wrote:

[...]

>
ℝ is what.we.mean.by the continuum because
ℚ is gapless (ℚ does not 'next')
⎛ ¬∃q,q″ ∈ ℚ: q < q″  ∧
⎝   ¬∃q′ ∈ ℚ: q < q′ < q″
and ℝ completes ℚ
⎛ ¬∃S ⊆ ℚ:  {} ≠ S ᵉᵃᶜʰ<ᵉᵃᶜʰ ℚ\S ≠ {}  ∧
⎝   ¬∃r ∈ ℝ\ℚ:  S ᵉᵃᶜʰ< r <ᵉᵃᶜʰ ℚ\S

I tried to simplify those expressions,
but I'm not convinced I got that right.
This is right:
⎛ ¬∃r,r″ ∈ ℝ: r < r″  ∧
⎝   ¬∃r′ ∈ ℝ: r < r′ < r″
⎛ ¬∃S ⊆ ℝ:  {} ≠ S ≠ ℝ  ∧
⎜  S ᵉᵃᶜʰ<ₑₓᵢₛₜₛ S ᵉᵃᶜʰ<ᵉᵃᶜʰ ℝ\S  ∧
⎝  ¬∃r ∈ ℝ:  S ᵉᵃᶜʰ< r ≤ᵉᵃᶜʰ ℝ\S
ℝ is gapless and complete.

such that
crossing ℝ×ℝ.curves intersect,
either in ℚ or in ℝ\ℚ

That is what I think is the motivation
for what.we.mean.by the continuum.