On 09/29/2024 12:02 PM, Jim Burns wrote:
On 9/26/2024 4:50 PM, Ross Finlayson wrote:
On 09/26/2024 06:48 AM, Jim Burns wrote:
On 9/25/2024 9:50 PM, Ross Finlayson wrote:
On 09/25/2024 02:00 PM, Jim Burns wrote:
On 9/25/2024 2:44 PM, Ross Finlayson wrote:
>
How would you define "atom"
the otherwise "infinitely-divisible"?
>
It's seems quite Aristotlean to be against atomism,
>
I am anti.atomized.ℝ (complete ordered field)
You seem to read that as anti.atomized.anything.
>
How do I make claims to you (RF) which
are only about the things I intend?
>
For example, how do I claim to you (RF),
for a right triangles but not for any triangle, that
⎛ the square of its longest side equals
⎝ the sum of the squares of its two other sides
?
>
I enjoy reading that, Jim, if I may be so familiar,
I enjoy reading that because it sounds _sincere_,
and, it reflects a "conscientiousness", given what
there is, given the milieu, then, for given the surrounds.
>
Thank you.
>
I am anti.atomized.ℝ (complete ordered field)
You seem to read that as anti.atomized.anything.
>
How do I make claims to you (RF) which
are only about the things I intend?
>
For example, how do I claim to you (RF),
for a right triangles but not for any triangle, that
⎛ the square of its longest side equals
⎝ the sum of the squares of its two other sides
?
>
>
A right triangle: either has one right angle,
or two angles that sum to a right angle.
The sum of the angles is the angle of the sum.
Then, as with regards to atoms, here we might recall
for example the notion of "the bullets", as with regards
to the contiguous, where at least one notion of the atoms,
is as of the extended bodies in uniform motion,
it's "placement, yet moving", from arbitrary A to B or 0 to 1.
This was mentioned about thirty years ago or when there
was a time when FredJefrries on the board, had, that
at some point, my goal was to convince FredJeffries,
that triangles have a well-defined area and it's bh/2.
How I went about that was for an axiomless geometry,
as that a spiral-space-filling curve in any dimensions,
is a line segment in one dimension, as with regards to
that in any dimensions, an abstract spiral-space-filling
curve, builds that what results is that distance, the
shortest distance, is a straight line, that the act
of drawing a line segment, or scribbling a dot, are
as of the same action, with regards to the inner,
and the outer, the space, of the points.
Then a square lattice gets built off that,
that a triangle is as after the affine of
that, the half of that. This basically has
that equipping via "axiomless geometry" an
equi-interpretable "Euclid's geometry", what
are otherwise undefined terms, like "the opus
of the drawing the pencil the line", as with
regards to geometry, and perspective and projection,
and motion, and geometry is motion and perspective
is projection.
So, before atomization, of a line into points,
and of course there are both ways why "lines are not points"
and "points are not lines", yet as well since antiquity
that "a plane is a regulus of non-crossing lines",
that "a volume from at least one perspective its projection
is a regulus of non-crossing planes", and so on ad infinitum,
about how from beneath it's arrived at, "a line is drawn of points",
and above, "space is all spaces".
This thusly introduces that perspective and projection,
are in effect part of geometry, in motion, rest and motion.
Then, where there's a theory with nothing in it, yet
natural integers, and some laws of arithmetic with
regards to merely the monotonic and constant and
increasing, the constant monotone strictly increasing,
then relating those, to a course-of-passage of line-drawing,
results line-reals and the differences their iota-values.
Now, as mentioned before, the arithmetic of these are
not as of both the usual sums and products, one or the
other gives with regards to iota-sums and iota-multiples,
that as courtesy division the iota-multiples are the
iota-values, while the iota-sums, are only as the
constant monotone strictly increasing.
Then, for the complete-ordered-field: it's _not_ members
of the completed ordered field, only defined as by its
extent [0,1], then density and completeness and measure,
1.0, why "line-reals" and "field-reals" are two _different_
models, of continuous domains.
Of course then such a construction to exist together with
others, mathematically, has to fall out of results like
the results for otherwise un-countability of 2^Aleph_0,
that the natural's cardinal is Aleph_0, and via the
composition of functions, may not be 1-1 and onto a bijection,
a domain in the cardinal 2^Aleph_0, where the definition
of the cardinal is the equivalence class of _Cartesian_,
specifically, functions transitively having 1-1 and onto
bijections among them, howsoever contrived, or as with
regards to Cantor-Bernstein that bijections exist or
Cantor-Schroeder-Bernstein that surjections or injections
either way suffice. So, this putative function "EF",
for "natural/unit Equivalency Function: cardinal equivalency of
naturals and [0,1], discrete domain, continuous range",
falls out uncontradicted, for the number-theoretic results
of otherwise uncountability, of continuous domains.
(There's also that the space of expressions in real functions,
has a higher cardinal than R the complete ordered field,
yet has nowhere else to cross except the linear continuum,
the zeros. This is usually called "the long line of duBois-Reymond,
who also discovered the anti-diagonal argument".)
So, line-reals and field-reals are altogether different _sets_,
and line-reals themselves are not established as via a _Cartesian_
function, which makes otherwise all such sorts contrivances for
Cantor-Schroeder-Bernstein theorem, instead that they only agree
about the bridges or ponts the lattice points, where first there
are naturals N, then there is [0,1] from N x omega in N, then
there is Q as N/N, then there is R as Q + least-upper-bounds.
Thusly, real-values are either integer part and non-integer
part the integer-value and iota-value, or, exclusively,
member of the equivalence classes of sequences that are Cauchy,
according to the universally agreed formalism of standard
real analysis, number-theory's and analysis' interest in
descriptive set theory.
(Then there's "extra-ordinary set theory" and "ubiquitous ordinals"
as with regards to the set-theoretic powerset uncountability,
with regards to set theory and ordering theory: two theories,
in each other.)
Then, it's nice and simply that as a non-Cartesian function,
this N/U EF, it's got its own definition of continuity,
it's got its own model of the space of all sequences 2^w
as a "square" Cantor space, it makes its own convergence criterion,
zero to one, and otherwise is the closest, simplest, function there is:
to a function with discrete domain and continuous range.
I have a mathematics degree so I made sure to internalize
these distinctions since a long time ago and they are
part of mathematics.
Then, it's about the most usual discourse since antiquity.
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?