Re: Incompleteness of Cantor's enumeration of the rational numbers (conserved infinity)

On 11/20/2024 07:43 PM, Chris M. Thomasson wrote:

On 11/20/2024 7:35 PM, Ross Finlayson wrote:

On 11/20/2024 07:05 PM, Chris M. Thomasson wrote:

On 11/20/2024 5:18 PM, Ross Finlayson wrote:

On 11/20/2024 05:00 PM, Chris M. Thomasson wrote:

On 11/20/2024 4:02 PM, Ross Finlayson wrote:

On 11/20/2024 12:23 PM, FromTheRafters wrote:

on 11/20/2024, Ross Finlayson supposed :

On 11/20/2024 12:05 PM, FromTheRafters wrote:

WM wrote on 11/20/2024 :

On 20.11.2024 15:14, FromTheRafters wrote:

WM formulated on Wednesday :

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It does not make it wrong, but it unmasks it at imprecise.
That's
why I don't like it. We can do better.

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It works well enough.

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Really? Then you can answer the following questions:
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Let every unit interval after a natural number on the real
axis be
coloured white with exception of the powers of 2 which are
coloured
black. Is it possible to shift the black intervals so that the
whole
real axis becomes black?

>
No, of course not.
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Or: Let every unit interval after a natural number on the real
axis be
coloured as above with exception of the intervals after the odd
prime
numbers which are coloured red. Is it possible to shift the red
intervals so that the whole real axis becomes red?

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No, of course not.
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What colour has the real axis after you have solved both tasks?

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Depending on the order of the tasks. I think half red or half
black.

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Well you have to reference academic reference and describe
"supertask"
besides "asymptotics" about where "the asymptotic density of
black or
red respectively is 1 in the limit", that you point to "supertask"
instead of mumbling like it's not already considered by proper
minds,
not just ketonic neck-flap gaspers of having failures altogether
of any sort of related-rates problems.
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This is mathematics: humor is irrelevant, and so is what
anybody "thinks", or, "feels".
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It only matters what "is", and there's a language of it,
so use it. (Or lose it.)
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Good sir

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If painted black and then red, it will be red. If painted red and
then
black, it will be black. These are real intervals, and as such I
assume
real powers of two. In both scenarios, none of the negative real
axis is
at all affected.

>
"Restricted Sequence Element Interchange" is an idea that
is a sort of "conservation principle" about things in an
Integer Continuum or Linear Continuum, here for example
an Integer Continuum. The idea is that any switch, as much
as it changes a plain 0101 to 0011, happens once-at-a-time
or the pair-wise, about basically, "after so much time given
to find an offset to exchange and another for its place,
and to update the state of the data structure that it is so,
that it's a matter of book-keeping and related-rates or
a system of algorithmic resources in numerical resources,
and time", that it's not merely giving x_infinity when
"at time 0 < Sum 1/n^2 < 1 that element n changes from
0 to 1" that at t_oo at n = oo that it's all 1's,
that it's so asymptotically, or that the density as
always filling in closer to the origin has that any
first different is arbitrarily far away, still has
that it's an honest account of book-keeping to make
that into a structure as if you had to implement it
and more than merely a lazy, forgetful mathematician's
exercise in induction that can easily arrive at
from 010101... to 00000... or 111111....
>
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Anyways there's a theory about these things that
basically make for cases besides those that just
shove off the end and put it off forever, besides
the "asymptotics" is what's called "supertasks".
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These may include for systems that are merely
"very, very large" when not "actually infinite",
that some practical or effective infinity, yet
results as a "point at infinity" which is a critical
or accumulation point, for the swapped-out items.
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Like a "point at infinity", a "prime at infinity". Or not,
it's among things entirely independent standard number
theory, which some have as that the integers don't actually
have a standard model anyways, only fragments and extensions.
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Anyways these sorts of things make for reasonings when
things exchange and conserve besides one-sidedly shove off.
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A prime at infinity? Keep in mind that there is an infinite number of
primes. So, are you talking about perspective as in a point at
infinity?

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What about it?
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Number theorists have various ways to define a point
at infinity, vis-a-vis geometry's usual notion as
of a perspective point-at-infinity, and projective
geometry's projective point-at-infinity, and number
theorist's compactification of the naturals or variously
with regards to the fundamental theorem of arithmetic,
whether it's so at infinity, or not.
>
It is rather regarded that these notions are _significant_
and _relevant_ and as well that they're _independent_,
usual enough fragments of theories of fragments of models
of numbers, or fixed views, and these kinds of things.
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So, there are models of integers with a "prime at infinity",
i.e. its only multiplicative factors are itself and 1,
and it's defined. There are others where it's composite,
for example being a product of each of the primes, there
are others, there are each the others. A "prime" at infinity
decides some things and makes an _opinion_, it's a _singular_
of what's a _multiplicity_ of models.
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It's usually said to be "number theory's" and no other
theories of mathematics thusly get any say at all about it,
unless said theory includes all the multiplicity of its
considerations. In this sense a usual nominalist fictionalist
"we can't say nothing" may be a good thing, as they can't say
anything wrong about it either, yet "you can't say nothing, either"
is considered objectionable, because number theorists can and do.
>
Consider double primes: two primes, separated by two, for
example, 11 and 13, or, 17 and 19. Now, there are many, many
less, of the double primes, than there are of primes. Then,
for triple primes, there's only 2, 3, 5. There are no other
(known) finite triple primes. Yet at infinity, it could
be in or among double, triple, even quadruple primes,
even n-tuple primes, in various models of integers.
>
So, anyways, the projective and perspective are very
relevant geometry beyond renderings, and about it.
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A point at infinity tends to be finite for the perspective?

>
There's nothing greater, so, no, not necessarily.

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Well, think of define a point at infinity for a game, say its doom or
something with an odd aspect ratio.
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Viewing the wide-open sky, it's full of perspectives, axially,
each a point at infinity, "un-rendered perspective infinity",
and each represents a projective point at infinity, because
there's what called a space inversion, between any local point
of reference, and it, the distance inverted as that inverting
a small distance with respect to a unit results a large distance,
and arbitrarily small and arbitrarily large.
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Think of it this way:
zero meters per second,
is infinity seconds per meter.
And vice-versa, ....
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So, having the projective point at infinity, in the _mental_
relation, extra the _sensual_ relation, the nous for the
pheonomenos, has the projective point at infinity is always
seen from what's considered an origin or perspective, as
a perspective point at infinity, not to be confused with
a prospective point at infinity, which it also is,
a projective point at infinity, where projection represents
action at a distance, as it were, or all sorts usual notions
of the perspective and projective usually from a theory of
graphical rendition and particularly of the realist sort.
>
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Or, both art and architecture's "point at infinity" may be
considered the same, while the camera's and the eye's actually
may be more as of about the optical effects of light really,
with regards to imaging and films and theories of light and
optics since Lucretius and Grosseteste, then particularly
Fresnel, yet including the ancient Sumerians, with regards
to what before light-pollution was the sky-survey: stars at night.
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Then mathematics includes those and what inductive inference
as it were never arrives at, a mathematical point at infinity,
actual, as it were.
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For example, in a theory where gravity's speed is infinite,
or the classical Newtonian theory, and one may aver that
Einstein's is "classical in the limit", has that a lazy
forgetful mathematician can imagine all the way to infinity
and back: in zero time.
>
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Then also there are a ton results about the compositionality
of elements algebraically and arithmetically in very large
finite algebras what are practically or effectively infinite,
with regards to a numerical concern, yet not actual, with
regards a numerical concern, that "prime (or not)" at infinity
basically predicates what happens in what else is a "supertask".
As demonstrated here, "beyond asymptotics" may include "supertasks",
including those whose outcome is never predicted by finitism.
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So it employs octrees and sprites in a usual sort of
otherwise ray-tracing way as to a projection according
to a perspective on a screen? Same difference.
Imagine a mini-map. Is it really a thing?
Mathematics is a big-boy theory.
Don't get me wrong, I've passed lots of games, and on hard.
Then ray-tracing is a remarkable thing, and after procedural
graphics are procedural 3D elements, quite often usually of
triangles, yet, "tesselations", that when we read Foley and
van Dam, and Graphics Gems and Game Developer and Gama Sutra,
and employ the MMX/XMM and later rasterizers, for raster/rake
imaging where it's not so much these days vector displays,
employing various matrix multiplications or often quaternions,
then as with regards to kinematics and procedural graphics
the pixel-filling as with regards to voxels, about specular
lighting and reflections, and natural-looking things, it does
seem a long way from simple 3D-looking flight simulators on
an 8-bit CPU, then there's something to be said for "Gran Turismo".
The mechanics of the thing is quite a thing besides the
graphics of a thing, then that it's to be kept in mind
that usual imaging on the screen like pictures and videos,
or moving pictures, are simply designed to escape most
people's perceptual differentiation, and besides that
most people are trained to "see the screen".
Some people can't even see the screen, or for example
can't even employ a usual laboratory microscope at all,
when for whatever reason through the viewfinder there's
nothing that comes in. Yet, at the same time, sometimes
one can look at the Moon, and see it in perfect resolution.
See, you're talking about a, "simulation", which is a
very interesting thing,
Simulated what exactly, ....
Yet, mathematics is a world that anybody can make
in their own mind.
And it's real, ....
So, yeah, mathematics is a big-boy theory,
and it's got infinity in it.
Then, physics, with imaging, is also analog.
How to represent this digitally, or "simulate",
is of course its own great enterprise.
Mostly though it involves symbolic representation,
besides rasterizing that.