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

On 9/20/2024 5:15 PM, Ross Finlayson wrote:

On 09/20/2024 12:26 PM, Jim Burns wrote:

On 9/20/2024 2:10 PM, WM wrote:

On 20.09.2024 19:51, Jim Burns wrote:

 

Put pencil to paper and
 draw two curves which cross.
There is a point at which
 the curves intersect.

 

Theorems or axioms?

 Here, a theorem.
⎛ (axiom)
⎜ The sets of ZFC exist.
⎜ (theorems)
⎜ ℕ exists
⎜ ℤ exists
⎜ ℚ exists
⎜ The set of Q.subsets
⎜ {S⊆ℚ:∅≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ\S≠∅}
⎜ exists and is the complete ordered field.
⎜ The Intermediate Value Theorem is true of
⎝ {S⊆ℚ:∅≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ\S≠∅}
 Here, an axiom.
⎛ (axiom)
⎜ The IVT is true of ordered field 𝔽ᑉᐧⁱᵛᵗ
⎜ (theorem)
⎝ 𝔽ᑉᐧⁱᵛᵗ is Dedekind.complete, and thus is ℝ
 

"Drawing" a line, "tire en regle", or curve,
has that when you put pencil to paper,
and draw a line, or curve if you will,
and life the pencil and put it back down,
and draw another one, intersecting the first:
the _curves_ cross.
>
... At a point, of for example where
they're incident, they coincide.

 Yes.
Because continuous curves must cross,
bounded nonempty set S must have a least.upper.bound.c