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

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

On 09/18/2024 01:51 PM, Ross Finlayson wrote:

On 09/18/2024 12:37 PM, Chris M. Thomasson wrote:

On 9/17/2024 7:58 PM, Ross Finlayson wrote:

On 09/17/2024 01:11 PM, Jim Burns wrote:

Unlike ℕ and ℤ,  ℚ and ℝ do not 'next'.

[...]

>
Put pencil to paper and draw a straight line,
each of the points were encountered in order.
No matter how fine it's sliced, ....

>
Sometimes called "Hilbert's Postulate of Continuity".
>
Which he says is required, ....

In a straight line or in a straight anything,
its order '<' is
⎛ connected (x≠y  ⇒  x<y ∨ y<x)
⎜ transitive (x<y ∧ y<z  ⇒  x<z)
⎜ asymmetric (x<y  ⇒  y≮x)
⎝ irreflexive (x≮x)
That states that
each of the points are encountered in order,
but
it leaves unstated what points are there,
and whether they are points which 'next'.
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
such that
crossing ℝ×ℝ.curves intersect,
either in ℚ or in ℝ\ℚ
We have found that it is necessary and sufficient
for crossing curves to intersect
in order to serve as what.we.mean.by continuum.