Re: Class/Set distinction, was Re: 2022-10-10

On 01/22/2024 06:49 PM, Ross Finlayson wrote:

You know I haven't read your library yet then for for drawing a circle,
it's generally appreciated that drawing a circle is difficult freehand,
while, a  line, or stroke, can sometimes be drawn in a faster line or
straighter,
or, sometimes the curve can make a round, where, the drawing the curve
is "The Turtle of the Logo Language", which is like a snake or otherwise
filling in neighboring regions according to parametrically that the
turtle
points in a direction and draws line of a length.
>
For drawing a circle I appreciate, that, parametrically for an angle
about
"2 Pi radians", sine and cosine fill in the +-1 quadrants, about some
origin
or co-site or "the pixel in the middle", in pixels and voxels or
picture elements
and volume elements, after ray-tracing in 3D what is drawing.
>
>
It has to be right from the center, about that there will be some
values of
rotation, that, should or will give, ellipses, while in the
course-of-passage
one rotation, "the circle of the same perimeter does not have this
ellipsoid".
>
>
I.e., for each angle offset, it has an apparent perimeter.  There's a
circle
with the same perimeter.  The ellipse has only two axes of symmetry,
X and Y.  They cross and that's the center of the ellipse and the two
nails that define the "tautochrone" what in constant time draws the
same perimeter, so, it's only an ellipse, if, it results the ratio of the
half-height and half-width, of the ellipse, must be same, as
>
>
root c, + b
   and
root c,  + a
>
of side b in x and a in y and side c the hypotenuse the bisection of
the obtuse angle of what results the triangle that has same perimeter,
as a polygon, as the length of the diameter, a line.
>
Then I forget the formula for perimeter of one quadrant or
"drawing chord", then to figure it with "secant".
>
So, this is what I am thinking is that according to some properties
I heard that defines an ellipse in terms of a circle instead of a conic
section, of course that besides it's defined in conic sections only
as "not parabolic, or circular, or, hyperbolic, ...", for making the
"cone"
as the usual projection, in that the cone is open in the direction of
travel or "vision" in the sense of drawing perspective.  (View, image.)
>
>
I figure that by computing the value of the perimeter of the ellipse
with
>
width:  2 (root c + a)
height: 2 (root c + b)
>
it indicates the circle with diameter 2b as being an ellipsoid of the
circle.
>
>
Then, there's that, at any point, in the first quadrant (where both
coordinates
are positive), for a point D on the circle, O the center and A the
point where OA the line is a,
there is a triangle with sides OD and AD, two "diameters" of a sort,
and OA - that with sides
a = OA, o = OD, d = AD, that a + o + d = 2(a + b + c).
>
Then, according to the angle between a and o, is to figure out how to
break the
obtuse triangle that is not isosceles, also to find the points where
it's isosceles.
>
Here by "break" it's that I want to compute, given angle ao, that is
basically between
the x axis and the line from the center to a point on the ellipse, to
compute the length
of o, or rather to solve for the value of o, given b, and a.
>
Then, ellipse-drawing would be similar your circle drawing approach,
over 2 Pi radians.
>
>
Here it seems like the approach to breaking the triangle OAD, is to
break the obtuse
triangle into a right triangle, and a smaller obtuse triangle with
what was left of AD
the hypotenuse and what was left of OD the middle length, then from
the height of
this triangle, to, break down the resulting leftovers , to compute the
length of OD given
the angle between a and o, and, a, and b.
>
>
"Seems we're all to read Seminaire de Geometrie Algebrique du Bois Maries
as for Dieudonne and Groethendieck, but my French is poor and accents
terrible. "
>
>
>
"Ellipses don't have a radius, ..., it's two diameters, ...".
>
"Frame contractions:  fields and couples", ....
>
"Lagrangians: I ate a zinc today"
>
"Lagrangians: I ate a zinc today, the Lagrangian held up the derivative"
>
-- "it is not a lemma"
>
>
Already gave all the answers.
>
>
Heh, surprise:  two surprises.
>
>
"That a full logic, is built up in so few terms,
from that an empty logic is not an empty logic."
>
>
Here "Hartogs is the Pope, not Russell, ..., who is also not the Pope".
>
"Is Hartogs even Catholic?"
>
"Russell, is not the Pope."
>
>
>
>
Hmm.  So, the ellipse, is having a parametric definition, in a and b.
So is the circle, where b = c.  This is a triangle, connecting as if two
nails in a board and a closed loop of string, were drawn about them an
ellipse.
>
It's that the perimeter is same what is confounding.
>
For as if any about around drawing the loop, it's the same perimeter.
>
>
I read this fact I think in some Ordinary Differential Equations work,
....
>
That the perimeter was same - I had already drawn the ellipse, then as
what I'd wonder how is the Bezier curve, what must be functional, when
everyone knows Bezier though nice, is not polygonal and geometric at all.
>
Yet, here only I am weakening when there is to be breaking the triangle,
breaking off small right triangles, until the triangle is exhausted,
to compute
the length of the distance between the center, and some point on the
curve,
according to the angle, given a and c.
>
The first is is pretty clear or a's, figuring, that, at for example
small angles,
then it would be a would be large the first, pointing out a good
feature that
in the method, if correspondingly the first triangle is specifically
measured
when also the angle is small, or rather 90 or 270 degrees, small from
that,
otherwise for example at 0 and Pi radians, a's triangle is small, and
the recursion
to compute each next triangle, then starts from that it's _under_
where a's
triangle intersected the triangle and the ray up from a, the same rather,
that the triangle under that is right, and, it's base is according to
the angle
of a thus cos a on the unit circle, thetriangle's hypotenuse is up AD,
then that
the next next is what's left above it, in OA OD AD.
>
>
>
>
Cantor was very enthusiastic that the centrality of uncountability,
was fundamental.
>
He had found several proofs to said effect.
>
He was a model aggrandizer.  I.e. that his other conjectures were
contradictory, or
underdefined when translated to the models of numbers, he dropped them.
>
>
>
>
Then for something like "Cantor's sets where CH and GCH are strong,
are rather mostly
like sets yet instead in cardinal theory", is about for making
cardinals besides a primary
theory, nudging it alongside the set theory that include "Not CH",
"Not GCH", ....
>
I.e. it's said Cantor's goal mostly was the continuum hypothesis, but
there are ordinals.
The notion of cardinal theory came after Cantor's convincing
uncountability theorems.
>
In cardinal theory sets share a rank or order in cardinals, next rank
is up, while
in ordinal theory sets share a rank or order in ordinals, next rank is
next.
>
I.e. ordinals don't share a rank, but,in "well-founded" set theory,
the "simplest"
ordinals, the initial ordinal has a cardinals, the ordinals happen to
share.
>
>
>
I only have ten or so books on Ordinary Differential Equations, and
boundary
value problems, here the idea is making the triangle into a strip or
strips and
perhaps building squares what fill acute or obtuse from an angle and
axis,
these few different patterns, for a >= 0 and b <= c.
>
Then it's like "the ellipse is the catenary outside and all around,
equilibriating
pressure, while outside is compression as between discs".
>
They must already have one or its parts but here I'm still looking at
this "2 (a + root c)",
and "2 (b + root c)".  Adding values and roots of other values, must
have a graphical
representation.  I.e. "the square with area c its side", and b being
the same value,
makes that where b = root c, about that c = b^2, then that according
to addition
it simplifies to 2 (x + x), then to take advantage of where power
terms have additive
terms like logarithms, in usual replacements and substition what
result a solution,
that "according to the graphical representation its transforms under
terms", build
generally a making for that each these expressions are also parameters
themselves,
i.e. then as to how in terms of a and b, and 2a + b + c, and a + o +
d, that about 2 (a + root c)
and 2(b + root c), that a graphical representation results.
>
That c = b^2, here is for that where c = 1, the radius, then it's
different than "whatever
is the diameter of the circle of the ellipsoid, c > 1", about whether
exists only c = b = 1,
or, that it's root c < c.
>
>
"Ellipses don't have a radius, ..., it's two diameters, ...".
>
>
That's like "there's somebody in the pub, if they drink, they forget".
>
Then there's "if you came here to drink to forget, please pay in
advance".
>
>
The primes-as-multisets-courtesy-arithmetic is pretty great,
multiplications are mutiset-unions, resulting the combined contents.
>
>
About density in primes and the modular, for arithmetic, I'm wondering
I suppose about density of primes, and density of small and larger
primes,
when what results terms in arithmetic, factor both sides the primes,
out of the distribution of their roots existing or factors, where, there
are for factors and "fundamental theorem of algebra", real roots,
then for "fundamental theorem of arithmetic", ..., is for probabilities
in primes, then also, guarantees filling, when it's sure so many primes,
are under some bounds.
>
>
>
>
>
>
https://mathworld.wolfram.com/InverseCurve.html
>
Hm, this says the Peaucellier inverser, makes a linkage of lines
connected
at points that is a linkage, which resuils a drawing machine of sorts,
as to the pantograph arm.
>
" The lituus is the locus of the point P moving such that the area of
a circular sector remains constant."
   -- https://mathworld.wolfram.com/Lituus.html
>
>
"The locus of the apex of a variable cone containing an ellipse fixed
in three-space
is a hyperbola through the foci of the ellipse. In addition, the locus
of the apex of
a cone containing that hyperbola is the original ellipse.
Furthermore, the eccentricities of the ellipse and hyperbola are
reciprocals. "
   -- https://mathworld.wolfram.com/Hyperbola.html
>
>
https://mathworld.wolfram.com/ReflectionProperty.html
>
More facts about ellipses, ....
>
"In 1882, Staude discovered a "thread" construction for an ellipsoid
analogous
to the taut pencil and string construction of the ellipse (Hilbert and
Cohn-Vossen 1999, pp. 19-22).
This construction makes use of a fixed framework consisting of an
ellipse and a hyperbola. "
  -- https://mathworld.wolfram.com/Ellipsoid.html
>
"The" "parametric equations of an ellipsoid", ..., here notice under
inverses are
their re-parameterizations, i.e. what are various parametric
definitions that result
the same objects according to geometry.
>
>
A deck of cards has, each card, has fourfold synmmetry, horiztonal,
vertical, and either way
diagonally.
>
So, if a deck of cards is split in half by one of its axes of
symmetry, there results two decks
of cards, that can be split again, and again, but, not again, into
decks of cards with at least
half the legible pip.
>
Because the pips are on the corners, they can only be split diagonally
once, besides that
horizontally and certically they can be split once, each, resulting 4
pips those split into two,
for 8 pips, making four decks of rectangular cards or eight decks of
triangular cards.
>
>
>
>
>
Here it's for that ellipses have two axes of symmetry except the
special case spheres, that,
those split the quadrants to make what rolls up in cones, what results
that how much the
cone is not a right cone, is the same the eccentricity, so
parameterized the same way as by
a which is not the eccentricity,
>
Also there's for circles at each corner of a triangle and their
ellipses with respect to each
circle's coordinates with respect to each other.
>
>
>
Here it looks "eccentricity" and "curvature" are defined about same.
>
https://en.wikipedia.org/wiki/Eccentricity_(mathematics)#Ellipses
https://en.wikipedia.org/wiki/Confocal_conic_sections
>
>
https://en.wikipedia.org/wiki/Ellipse#Pins-and-string_method
>
>
So, here "a" is just the distance from the center to a locus.
>
>
>
Ellipse is two diameters, ....
>
>
>
About that OA is longer than OD when x < a, and up to 2a, , is for
that OD OA < c^2.
>
I.e., here the point is to work down, the part before and after, here
is that drawing the
first quadrant of this ellipse, sees that OA is the hypotenuse or
"long side", "the long side's
always the hypotenuse".  OA is only the long side up to point at (
a/2, f(a2) ), where f is some
function that's parameterized in part by a, and later b or c.  Then in
the rest of the curve OD
is the long sideas where it reaches  (a + 2 (a + root c), 0 ), that
f(a + 2(a + root c) ) = 0, or here
for example about f ( 3a + root c ) = 0.
>
>
  ---    "Gray-scale area-averaging"?
>
Mostly for sure there is that after something is already rasterized,
in a quadrant,
that basically affine transformations are computed, for all right
angles, just make
"8 orientations", ..., about off-by-one.
>
High screen resolution's afforded a lot of complacency in raster
registration,
and usual enough aliasing and artifacts, ..., of ... aliasing.
>
https://docs.oracle.com/javase/7/docs/api/java/awt/image/AreaAveragingScaleFilter.html
>
>
Have I implemented such before or an area-averaging filter, yes, so,
in the generation of
grayscale thmubnails from page images.  I invented and implemented one.
>
>
>
As they pass each other seeing each other their field of view takes in
both origins and destinations.
>
>
>
That's what you get for getting led along.
>
>
Here an example is "division by zero".  All these people starting with
zero and numbers,
then adding division, leaving out zero instead of no zero, these are
clumsy and cumbersome.
>
Not contradicting each other while still have mutually compatible
interpretations of views,
what result abstractly the value of fairness where it's the only
principle, that all the
definitions are composed into one definition, besides that inference
exists.
>
One of the other here for example "numbers with zero" and "numbers and
division",
entirely are differently varied smaller collections of direct
resulting inferences,
that only one or the other is eventually exhausted in any sense of
completion.
>
>
>
"Infinite" ordinals, ....
>
>
>
https://www.maa.org/press/maa-reviews/a-history-of-japanese-mathematics
>
"... and Wada Nei's contributions to the understanding of hypotrochoids".
>
"Sections of an elliptic wedge, for example, ...".
>
>
>
How about Tableau?
>
>
>
Various Goldbach conjectures are undecide-able in extra-Archimedean
extensions of integers.
>
So, the "don't care" bit was "already got a wider world".
>
There's also models with a triple prime at infinity.
>
Of course, that's in a number theory, with a prime at infinity.  There
are others, ....
>
Anyways though it's great youre boring and under-informed.
>
>
Saying it's in Windows doesn't mean being used -
most of that (..., user-space) uses _you_, not the other way around.
>
Old "ball and chain" as it were.
>
>
>
>
f = f'
  ...
>
f' = 0
f' = c
>
>
d^2  f = 0
/dx^2
>
   f = cx
>
>
I guess that depends whether it's me or your profile.  Like a
sock-puppet.
>
>
"Ordinary Integral Equations"
>
"Scholze and Clausen say they have already found simpler, ‘condensed’
proofs
of a number of profound geometry facts, and that they can now prove
theorems
that were previously unknown. They have not yet made these public."
>
"Relies on facts from stable homotopy theory, ...".  Adding a
univalency axiom is a lot of
strength, though, there are derivable immediately usual contradictions
of the illative
to the well-founded.
>
Differential equations:
>
f' = f
>
   e^x
   f(n/d), d->oo, n-> d
>
>
"Most special functions are solutions of differential equations, ...".
>
https://leanprover-community.github.io/mathlib_docs/measure_theory/measurable_space.html
>
>
https://leanprover-community.github.io/mathlib_docs/meta/univs.html
https://leanprover-community.github.io/mathlib_docs/logic/small.html
>
https://leanprover-community.github.io/mathlib_docs/model_theory/definability.html
>
>
Yeah, running out "not definable" helps remind that adding univalency
to ordinary
set theory, makes for that "Russell is not the Pope".
>
https://leanprover-community.github.io/mathlib_docs/set_theory/zfc/basic.html
>
>
Lean "Class":  "set Set".
>
https://leanprover-community.github.io/mathlib_docs/init/data/set.html#set
>
>
You see Lean defines Set in terms of membership predicates, but,
the only relation in ZFC is "elt".
>
Yeah, it's a usual hypocritical mish-mash, but, I suppose lots of
nCatLabs has
ready reading into anything, it..., "says".
>
https://leanprover-community.github.io/mathlib_docs/set_theory/zfc/basic.html#Class
>
>
"Class:  the collection of all classes".
>
Class-set distinction and the group-noun game,
makes for writing many simply inconsistent multiplicities in Lean's
words,
but if you ignore the part that disagrees with you then your ignorance
will be invincible.
>
Yes it looks a very usual these days' "the strength of ZFC and two
large cardinals".
>
https://leanprover-community.github.io/mathlib_docs/topology/subset_properties.html
>
>
>
>
Big ripoff of nCatLabs?
>

>