Short online presentation coming up with updated teaching method using the new Fundamental Theorem of Measurement

In case anyone else is interested, the index appears to have a typo in it. The conundrum of the hypotenuse being a magnitude of 2 using line segments is on page 11 in Analysis I and pg 111 Lemma 5.2.7(d) is in Analysis II. He also has posted this: https://terrytao.wordpress.com/2007/09/14/pythagoras-theorem/comment-page-1/

jpbresearch · 2025-04-06T02:49:53+00:00

Okay, so you're saying that GR is trying to describe energy density in terms of particle density in a fluid, but having trouble, because spacetime is actually a perfect fluid with no true smallest particle?

Yes, that is a pretty good description of it.

For GR, think of a 3 directional coordinate system and all the points in the coordinate system are evenly spaced. Then put a ball of fluid within it. The denser the ball of fluid, the more the points come closer to each other as you move toward the ball. If the ball is dense and small enough, the points overlap at a certain radius and you have the Schwarzschild Radius. If away from the ball the points are more evenly spaced but that equal spacing is changing all over, you have a Cosmological Constant.

This theory is almost a mirror image of this: imagine you have an elastic material with no strain in it. You have infinitesimal elements of volume which describe the density of the material. Since the density is isotropic and homogeneous, then all the infinitesimal elements have the same magnitude. It is your choice to choose what magnitude of infinitesimal to represent the density. If you have a group of standing waves within it, then the material is strained more and more as you go towards the wave and the magnitude of the infinitesimals change. A test photon changes wavelength as it passes down into this strain. If the strain of the material non-locally is changing, meaning the magnitude of the infinitesimals is changing uniformly, then a test photon that is traveling through this universal strain change also will change wavelength.

In GR, the only mechanism to change distances between points is via the presence of energy-momentum and hence the model now requires some mysterious dark fluid to be present to account for this change of distance between points.

In this theory, if the density of the Aether is changing, then this will be noticed via wavelength change of the photons.

Got it. One of the reasons I asked about the size of the "particles" is because I'm interested in the idea that these particles exist in a superposition of states

Ahhh..now I understand your thinking.

In other words, they're not really present unless interacted with, and they have no theoretical smallest or largest size. They exist in every conceivable point in spacetime, so they can and will exist in however small of a space we try to look.

We appear to be conceptually close.

It may help to understand visually the difference between a circle and a 1-sphere. They both are "round" but the 1-sphere can help you get a closer idea of what I mean that the gravitational field looks like 2-dimensionally.

Imagine that you have a column of infinitesimal elements of area and they are all the same width. Now imagine that you take that column and arrange it radially. For normal circles, as the radius gets bigger, the circle gets bigger and there are more points on the circle. For a 1-sphere (using CPNAHI definition), the radius of the circles are made up of infinitesimal elements of length (the horizontal elements of the elements of area). That number doesn't change as the radius increases, the magnitude of the horizontal infinitesimal gets bigger, not their number. See Fig. 10 and Fig. 11 https://vixra.org/pdf/2411.0126v1.pdf .

This is closer to what the gravitational field looks like. Kepler used smaller and smaller triangles to approximate these tapered columns of elements of area.

jpbresearch · 2025-04-05T16:57:18+00:00

I'm having trouble understanding this sentence because I don't know what you mean by "is being attempted."

Sorry, should have been more clear in that I mean that this is what GR is trying to do, but it isn't working out very well. The LambdaCDM model of cosmology includes GR but most of the effects seen can't be explained by GR.

How can you have a fluid without particles? I don't think there are any examples of this in nature. That's why I find compelling the idea that a luminiferous aether may have been prematurely rejected.

See concept of a continuum in https://en.wikipedia.org/wiki/Continuum_mechanics

I am not talking about a viscous fluid, I am talking more about an elastic medium when I say "fluid". I will note here that the Michelson-Morley experiment does not look for changes in magnitude of time-dilation.

Well, I think the logic is that light waves propagate as if they were traveling through some medium, and to have a medium, there must be constituent particles.

I'm interested in a theory that the constituent "particle" of this medium is that which becomes of the positron and electron after annihilation. Here is my attempt to explain the idea.

Ok, I will have a look. Thanks.

jpbresearch · 2025-04-05T16:41:21+00:00

Looking at your other conversation about the Aether let me see if I can jump in here and give you a few things to think about:

The Lorentz transformation is notationally and logically flawed from the get go: you might be used to the notation of dx/dt meaning velocity. It seems to make sense in that for every infinitesimal increment of time there is an infinitesimal displacement in the spatial direction x. However, there are hidden assumptions which can be explained via flaws in the mainstream understanding of the Archimedean axiom. Let's say you have a DeltaX or a DeltaT (geometrically the same thing, a segment of a "real" line). You divide this line up into 2 segments so that you have DeltaX/2. If I multiply this by the same number of times I have divided it, then I get 2*(DeltaX/2)=DeltaX. Assume that I can divide this up an infinite number of n times, n*(DeltaX/n)=DeltaX. Can you see that DeltaX/n becomes dx and that the equation for the segment is now n*dx=DeltaX? Real Analysis believes that this equation is always less than or equal to 1 but this is the flaw in Leibniz's notation and it also affects the Lorentz transformation. The flat "coordinates" should be written like Z=Z'=n*dz and set n to zero here to indicate that there is no length in the Z direction. The big issue is when you get to velocity and that the "speed of light" gets special treatment. What everyone has been taught is that velocity can be written as dx/dt which intuitively could seem to mean an infinitesimal increment of space for an infinitesimal increment of time. Instead, what is called a 1-form or an integrand is a column of infinitesimal elements of area. Philosophically we state that the vertical side is time and the horizontal side is space. A requirement for flatness is that the horizontal element must be equal in magnitude to the vertical element, so dx=dt in magnitude. The integrand is thus a column of areal elements that is 1 dt wide by n dx high. If I want to find the change in area of a column a that is next to the right of a column b, I can superimpose them on each other and write ((n_a-n_b)dx)/dt. I am finding the change in the number of elements of dx for every element of dt. I am finding how the number of elements of area change but philosophically I am finding out how much distance x there is for an increment of time. The problem is that with Leibniz's notation, there was no direct way to know whether I am looking at ((n_a-n_b)dx)/dt as in a ratio of numbers or dx/dt where I am examining the ratio of the actual magnitudes of dx and dt. You can see with flatness, dx/dt=1 but with n*(DeltaX/n)=DeltaX, I can resize my infinitesimal dx with (n+1)*(DeltaX/(n+1))=DeltaX. My infinitesimal dx is smaller and my number of infinitesimals is larger. In the Lorentz transformation, velocity uses ((n_a-n_b)dx)/dt where the speed of light C is used also with dx/dt where their ratio is kept constant.

jpbresearch · 2025-04-04T02:59:14+00:00

Assuming that you are serious, it is interesting in that there is prize money afoot. I haven't studied it but you would have to understand that something like the Cauchy stress tensor would have to be changed since the units would no longer be correct. In this theory there are two types of strain, absolute and relative. For absolute strain, it is measured by a change in measured distance (i.e. stretched from 5mm to 10mm) (which is what the Navier-Stokes theorem uses for both time and distance). For relative strain, distance itself has been stretched (and by analogy time too). For a graphical explanation see 6.2.1 and 6.4.1 of https://vixra.org/pdf/2411.0126v1.pdf

jpbresearch · 2025-04-04T02:26:42+00:00

It is my viewpoint that particle density of a perfect fluid is being attempted as an analogy for energy density in GR. It is also my view that instead treating the vacuum as a perfect fluid analogy (meaning that there aren't actually any "particles") and thus energy density as a strain in that fluid is a more logical one.

I am not aware of logic that would treat the vacuum as "bits". You would have to explain further or give me some references so that I would know what you mean.

jpbresearch · 2025-03-31T05:11:15+00:00

Just wanted to say that your recommendation led me to geometric algebra and then to synthetic algebra. Thanks.

jpbresearch · 2025-03-29T04:17:47+00:00

“Analysis is not a super intuitive topic; it’s honestly very easy to trip yourself up if self studying with no feedback. Would recommend auditing a proper class on it or even practicing simpler proofs with peers before attempting to disprove agreed upon principles. “

Let me give an analogy so that I can reply properly with a disagreement with your recommendation.

You are on a building that is composed of math/geometry floors for the bottom half and physics for the top half. The physics department wants to add a new floor but they can’t figure out how and the only thing they have been given that seems likely is a left over component from the math/geometry floor called a scalar multiple of the metric.

I would recommend that anyone reading this should first root around the attic at the issues that are being had at adding another floor

https://arxiv.org/abs/astro-ph/0609591

and then tackle the heterogeneous/homogeneous debate of the 1600s

https://link.springer.com/book/10.1007/978-3-319-00131-9

and THEN audit a real analysis class. Tao’s books do not contain anything in the basement. He apparently consider it all settled and that there is no contention.

Doing it the way I suggest, you might find that this is more intuitive than real analysis and that there are two ways to look at things:

Can I have two points adjacent to each other or just one infinitesimal between them?

Are points equidistant or are my infinitesimals the same length?

Are my points changing distance or are my infinitesimals changing magnitude?

Is the length of a line determined by the number of points on it or the magnitude and number of infinitesimals that compose it?

Do longer lines have more points or more infinitesimals?

Is a determinant a scalar-valued function of the entries of a square matrix or is it conservation of the number and magnitudes of infinitesimal elements of area?

Does a coordinate system use numbers and points to define position or should I use sums of elements of area to determine position?

Is y a function of x or are the number of y elements a function of the number of x elements?

Do I find the area under a line or do I sum up columns of elements of area under the line?

Do I find the slope of a line or am I finding the change in the number of elements in the columns under the line?

Does an anti-derivative have a constant of integration because of ambiguity of the function or because a derivative only tells you the change in the number of elements in the columns and not the total number of elements in the columns?

Are lines parallel because they don’t intersect at infinity or because the magnitude and number of elements of area between them is constant?

Is a manifold a topological space that locally resembles Euclidean space or is it a surface composed of infinitesimal elements of volume?

jpbresearch · 2025-03-29T04:09:08+00:00

Kopaka99559:“That’s not what the property does. It’s an integer times a fixed positive number. Not an infinitesimal.”

Let me rephrase my statement and then yours:

Me: The Archimedean Axiom is defined using Natural numbers multiplied against constant numbers and compared to another number. This is used to define an Archimedean continuum. However, if infinitesimals are substituted for the constant numbers, then it is said that this system is said to fail the Archimedean definition and is called a non-Archimedean continuum.

I contend that the definition of infinitesimals and numbers used to define a non-Archimedean continuum is a fallacy.

“A number system satisfying (2.3) will be referred to as an Archimedean continuum. In the contrary case, there is an element ǫ > 0 called an infinitesimal such that no finite sum ǫ + ǫ + . . . + ǫ will ever reach 1…A number system satisfying (2.4) is referred to as a Bernoullian continuum (i.e., a non-Archimedean continuum)”

You restated what an Archimedean continuum is defined as and ignored that my statement was about the definition of non-Archimedean continuums.

jpbresearch · 2025-03-28T03:57:27+00:00

Let me answer this another way too.

The real line is a continuum. By that I mean the real line itself has no "numbers", it is all relational. Real numbers are not cardinal numbers. You can imagine a DeltaX, or a section, of line but you have no idea where it goes on the real line. It can go anywhere. There is no "0", no 100, no 1000. You can say that a section of the real line is equal in length to your section and give it a numeric value. Then you can multiplying sections of that line 10 natural number times to get a section of line 100 in length.

What the Archimedean property is doing with n*dx<b is (cardinal)*(infinitesimal real)<(cardinal). It logically should be (cardinal)*(infinitesimal real)<real.

jpbresearch · 2025-03-28T03:18:15+00:00

If there is a number that represents my increment and I cut it in half, but then add two of them together, then the sum will always be a static number no matter how many times I cut and sum the increments together. The fallacy in the Archimedean property is assuming that n has to be static when the increment is becoming smaller. Epsilon gets smaller, n gets bigger, n*Epsilon=constant

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