Re: Does the number of nines increase? (axiomatizing completeness)

On 07/04/2024 09:45 AM, Ross Finlayson wrote:

On 07/03/2024 12:44 PM, Ross Finlayson wrote:

On 07/02/2024 09:02 PM, Jim Burns wrote:

On 7/2/2024 9:49 PM, Ross Finlayson wrote:

On 07/02/2024 06:06 PM, Jim Burns wrote:

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[...]

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With the least-upper-bound property for
reals in their _normal_ ordering and
reals in their _reverse_ ordering,
doesn't that sort of confound
just the usual partitioning scheme?

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For each nonempty bounded.below set S of reals
there is nonempty bounded.above -1⋅ᴬS
with a least.upper.bound -1⋅σ
σ is the greatest.lower.bound of
nonempty bounded.below S
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re: lub glb
You pays yer money and you takes yer choice.
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That is to say,
isn't any real number defined both ways?
Aren't they, neighbors? No different?

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No different. No problem.
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Reading more from Hermann in that podcast, gets into
that mathematicians and physicists sort of need to
get together, and, mathematics _owes_ physics.

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Even more obviously,
physics _owes_ mathematics, too.
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There's enough owing to go around.
See also: shoulders of giants
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Maclaurin wrote a great formalization of the
integral calculus in the time of Newton.
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Mathematics _owes_ physics
more and better mathematics of continuity and infinity,
here about continuous domains that physics has built
about Jordan Measure and Dirichlet Function,
and after complete metrizing ultrafilters,
these days including the quasi-invariant measure theory,
and the models that physics has,
of these continuous domains,
would suffer your formalist wrath,
because they model line-continuity and signal-continuity.
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(From mathematics, for physics.)
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So, party A models least-upper-bound as a reverse
ordering as from above, while party B models least-upper-bound
as a normal ordering from below. Do they meet?
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Can't they meet?
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MacLaurin and Gregory:  two of Newton's greater foot-stools.
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Also Coates, quite a great differential geometer.
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Line-drawing is really a pretty great concept,
it's un-defined in Euclidean geometry,
Hilbert makes it a postulate of continuity,
Aristotle has it as the contiguous continuum,
out in all real dimensions it makes a spiral space-filling curve,
Zeno requires it for getting from point to point,
it exhibits least-upper-bound so the field-reals can have one,
it's an example of multiple probability distributions of
probability distributions with multiple functions non-standardly
defining them, it's among functions who are their own anti-derivative,
what with a bit of only-diagonal non-Cartesian apologetics
inside set theory,
it makes Jordan measure,
and fulfills most people's notions of empty/full clock arithmetic.
Line-drawing: now in remedial.