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

On 9/17/2024 4:16 PM, Ross Finlayson wrote:

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

On 9/17/2024 2:57 PM, Ross Finlayson wrote:

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

Then, for initial segments or n-sets of naturals,
the LUB of {f{n < m)} is "next": f(m+1).

Yes.
(Presumably, you mean lub.{n:n≤m} = m. f()=? )
You might enjoy this:
⎛ Define ℕ as well.ordered and nexted.
⎜ well.ordered (A ⊆ ℕ holds min.A or is empty)
⎝ nexted (m ∈ ℕ has m+1 m-1 next, except 0=min.ℕ)
⎛ In a finite order,
⎝ each nonempty subset is 2.ended.
Consider upper.bounded nonempty A ⊆ ℕ and
its set UB[A] ⊆ ℕ of upper.bounds
A ᵉᵃᶜʰ≤ᵉᵃᶜʰ UB[A]
A is upper.bounded.
UB[A] ⊆ ℕ is nonempty.
UB[A] holds min.UB[A]
A holds min.UB[A]
Otherwise,
(min.UB[A])-1 is a less.than.least upper.bound
(that is, what.it.is is gibberish)
A holds min.UB[A] which upper.bounds A
min.UB[A] = max.A
Upper.bounded nonempty A holds max.A
Upper.bounded nonempty A ⊆ ℕ holds min.A
(well.order)
Upper.bounded nonempty A is 2.ended.
And, similarly,
each (also.bounded) nonempty S ⊆ A is 2.ended.
Upper.bounded nonempty A ⊆ ℕ is finite,
because
ℕ is well.ordered and nexted.