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

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

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

On 09/17/2024 10:59 AM, Jim Burns wrote:

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[...]

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I do seem to recall that
your account was around when it was set out,
for example,
that least-upper-bound is nice neat trivial next,

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In ℕ and ℤ
there are integers next to each other,
which is to say,
there are integers with no other integers between.
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In ℚ
there are no rationals next to each other,
because
there are no rationals with no rationals between.
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In ℝ
there are no reals next to each other,
because,
however close |x-y| = d > 0 is,
there are no d.sized gaps in the rationals,
so there must be rationals between.
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The least.upper.bound of (0,1) is 1
but 1 isn't next to any element of (0,1)
>
Unlike ℕ and ℤ,  ℚ and ℝ do not 'next'.
>
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Actually it's that really it's the "distinct/continuous"
together at all as of the "finite/infinite" the
"infinitely-DIVISIBLE", the, "INFINITELY-divisible",
where basically also there's that increment and division
are put together instead of the usual Peano/Presburger
so that the field is built in the middle as of two groups
instead of as through a ring.
Whatever works, right, ....
Then, for initial segments or n-sets of naturals,
the LUB of {f{n < m)} is "next": f(m+1).
It _is_ the least-upper-bound, ....