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

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 03:20 PM, Jim Burns wrote:

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:

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

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

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Yes.
(Presumably, you mean lub.{n:n≤m} = m. f()=? )
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You might enjoy this:
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⎛ 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)
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⎛ In a finite order,
⎝ each nonempty subset is 2.ended.
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Consider upper.bounded nonempty A ⊆ ℕ and
its set UB[A] ⊆ ℕ of upper.bounds
A ᵉᵃᶜʰ≤ᵉᵃᶜʰ UB[A]
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A is upper.bounded.
UB[A] ⊆ ℕ is nonempty.
UB[A] holds min.UB[A]
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A holds min.UB[A]
Otherwise,
(min.UB[A])-1 is a less.than.least upper.bound
(that is, what.it.is is gibberish)
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A holds min.UB[A] which upper.bounds A
min.UB[A] = max.A
Upper.bounded nonempty A holds max.A
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Upper.bounded nonempty A ⊆ ℕ holds min.A
(well.order)
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Upper.bounded nonempty A is 2.ended.
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And, similarly,
each (also.bounded) nonempty S ⊆ A is 2.ended.
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Upper.bounded nonempty A ⊆ ℕ is finite,
because
ℕ is well.ordered and nexted.
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Yet, didn't you just reject, "infinite middle"?
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Mid point:
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p0 = (-1, 0)
p1 = (1, 1)
pdif = p1 - p0
pmid = p0 + pdif / 2
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Put pencil to paper and draw a straight line,
each of the points were encountered in order.
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No matter how fine it's sliced, ....
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