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

On 9/20/2024 7:42 PM, Ross Finlayson wrote:

On 09/20/2024 02:38 PM, Chris M. Thomasson wrote:

On 9/20/2024 2:20 PM, Ross Finlayson wrote:

On 09/20/2024 02:15 PM, Ross Finlayson wrote:

On 09/20/2024 12:26 PM, Jim Burns wrote:

On 9/20/2024 2:10 PM, WM wrote:

On 20.09.2024 19:51, Jim Burns wrote:

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Put pencil to paper and draw two curves which cross.
There is a point at which the curves intersect.

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This proves that no line has gaps.

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A point (hypothetically) next to 0
has an absence of points between it and 0
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No such absence of points exists.
No point which is next to 0 exists.
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Theorems or axioms?
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In the physics, think on your wave/particle duality,
and the extended body, for example the wave-packet.
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"Drawing" a line, "tire en regle", or curve,
has that when you put pencil to paper,
and draw a line, or curve if you will,
and life the pencil and put it back down,
and draw another one, intersecting the first:
the _curves_ cross.
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... At a point, of for example where
they're incident, they coincide.
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Then these lines-reals these iota-values
are about the only "standard infinitesimals"
there are: with extent you observe, density
you observe, least-upper-bound as trivial,
and measure as assigned, length assignment.
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Consider for example minutes between 1:00 and 3:00
in this https://www.youtube.com/watch?v=fIexFF91Jl8
"Moment and Motion:  medical imaging technology".
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What it declares is that "complementary duals" have
that the points and the space and the space and the
points are for each other.
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Actually, each frame in this is part of a 3d process for constructing a
volume that can be used in a volumetric renderer. It is close to medical
imaging wrt DICOM:
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https://youtu.be/k9qpHcfiDho
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 I sure had a head cold that day.
 Studying medical imaging that way was kind
of totem-ish, like, maybe if I study medical
imaging today, then sympathetic magic will
help kick my cold and diseases begin with 'C'.
Seemed helped, .... Yet, that's humor for you.
  About "points-in-a-line", the classical notion
is "beads-on-a-string", here as with regards
to "the course-of-passage", that ordinals have
a course of passage through ordinals, and that
line-drawing, has a course of passage through
points.
 Divising a specialization of "in", "on", and
"about", the line, its points, helps to characterize
that line-drawing makes points "in" a line, that
zooming in via field operations points to, that
it establishes a vector field with all the arrows
reducing in region until their heads touch and
that they point-to, a point, "on" a line, then
as with regards to a super-fine comb each tooth
"about" the line.
 DICOM is a family of standards in medical imaging
data and metadata, sort of like JPEG for doctors.
See also "HL7". The JPEG2000 is pretty great,
I studied data formats a lot.

A single DICOM file can contain the image stack that its based on, right? Fwiw, here is a render of some of my experiments:
https://www.facebook.com/photo/?fbid=334035991088739&set=a.110008616824812
A screenshot:
https://i.ibb.co/jGjycbN/image.png
Strange results for sure.

  Chaos theory and its terms is sort of folded into
"dynamical modeling", as with regards to for example
"limit cycles" for "attractors", then as with regards
to for example Nayfeh's tome on non-linear oscillations.
  Anyways, here line-reals and signal-reals "exist"
next to field-reals, at least three models of
continuous domains, each with their own definition
of "continuous", all satisfying the same properties
of extent, density, completeness, and measure, from
which can be established most all their combined
character, as "real-valued".