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

On 09/18/2024 07:02 AM, Jim Burns wrote:

On 9/17/2024 10:58 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'.

>

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()=? )
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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.

>
Yet,
didn't you just reject, "infinite middle"?

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Please include in your posts
more of what you must consider
goes without saying;
you must, as you post without saying it.
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I can't track you without footprints
or candy wrappers or something
to hint at where you're going (metaphorically).
>
----
Yes,
I am still recognizing the falsehood of
"infinite middle" (0:ω symmetry).
The post to which you respond is pretty much
a Festshrift to rejecting infinite middle.
I cannot guess why you need to hear that.
And I have tried.
>
>
ℕ is uniformly well.ordered and nexted
  apart from 0=min.ℕ
That is what ℕ is.
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 From the uniform well.ordering and nexting
it is derivable (see prev.) that,
uniformly, finitely.many are before and,
uniformly, infinitely many are after.
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ℕ is uniformly on the near side of
infinite not.the.middle.
ℕ is lacking an infinite middle.
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>
>
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Yet, in the infinite limit, d -> infinity.
Please consider this as a tendered reserve and
not so much a purposeful omission of any sort,
where as with regards to my tens of thousands
of posts and brief essays to usenet there are
more than enough crumbs to go around.