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:

[...]

>
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, ....

Put pencil to paper and draw two curves which cross.
There is a point at which the curves intersect.
That is the Intermediate Value Theorem,
which can prove and from which can be proved
the Least Upper Bound Property.
That is what.we.mean.by line.
⎛ The line ℝ includes rationals ℚ
⎜
⎜ For each split S of ℚ,
⎜ if there isn't a rational qₛ at S
⎜ then there is an irrational rₛ at S
⎜
⎜ Each rational and irrational is at a split.
⎝
⎛ ℝ z ℚ
⎜
⎜ ∀S ⊆ ℚ: {} ≠ S ᵉᵃᶜʰ<ᵉᵃᶜʰ ℚ\S ≠ {}  ⇒
⎜⎛ ¬∃qₛ ∈ ℚ:   S ᵉᵃᶜʰ≤ qₛ ≤ᵉᵃᶜʰ ℚ\S  ⇒
⎜⎝ ∃rₛ ∈ ∁ℚ:  S ᵉᵃᶜʰ< rₛ <ᵉᵃᶜʰ ℚ\S
⎜
⎜ ℝ = ℚ ∪ ∁ℚ
⎜
⎜ ∀r ∈ ℝ: ∃Sᵣ ⊆ Q:
⎝ {} ≠ Sᵣ ᵉᵃᶜʰ< r ≤ᵉᵃᶜʰ ℚ\Sᵣ ≠ {}