# pengkuan

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193

1. ## Twin paradox when Earth is the moving frame

General equation for Space-Time geodesics and orbit equation in relativistic gravity 1. Orbit equation and orbital precession General Relativity explains gravity as Space-Time curvature and orbits of planets as geodesics of curved Space-Time. However, this concept is extremely hard to understand and geodesics hard to compute. If we can find an analytical orbit equation for planets like Newtonian orbit equation, relativistic gravity will become intuitive and straightforward so that most people can understand. From gravitational force and acceleration, I have derived the analytical orbit equation for relativistic gravity which is equation (1). Below I will explain the derivation of this equation. Albert Einstein had correctly predicted the orbital precession of planet Mercury which had definitively validated General Relativity. Equation (2) is the angle of orbital precession that this orbit equation gives, which is identical to the one Albert Einstein had given [1][2]. If this orbit equation gave the same result than Space-Time geodesics, then everyone can compute the orbit of any object in gravitational field which obeys General Relativity using personal computer rather than big or super computer. Also, everyone can see how gravity leads to Space-Time curvature without the need of knowing Einstein tensor. The derivation of the orbit equation is rather tedious and lengthy. So, for clarity of the reasoning and explanation, I have collected all the mathematical equations in the last section “Derivation of equations”, in which full details are provided to help readers for checking the validity of my mathematics. 2. Relativistic dynamics a) Velocity in local frame Take an attracting body of mass M around which orbits a small body of mass m, see Figure 1. We work with a polar coordinate system of which the body M sits at the origin. The position of the body m with respect to M is specified by the radial position vector r, of which the magnitude is r and the polar angle is q. Let the frame of reference “frame_m” be an inertial fame that instantaneously moves with m. Frame_m is the proper frame of m where the velocity of m is 0. So, Newton’s laws apply in this frame. Let am be the acceleration vector of m in frame_m and the inertial force of m is m·am, see equation (3). The gravitational force on m is given by equation (4). Equating (4) with (3), we get equation (5), the proper acceleration of m caused by gravitational force in frame_m. Let “frame_l” be the local frame of reference in which M is stationary. In frame_l m is under the effect of gravity of M, the velocity vector of m is vl and the acceleration of m is a l. As frame_m moves with m, it moves at the velocity vl in frame_l. … Figures and equations are in the pdf below: General equation for Space-Time geodesics and orbit equation in relativistic gravity.pdf
2. ## Twin paradox when Earth is the moving frame

We analyze the mathematical mechanism that slows the time of the traveler in the twin paradox and explain what distinguishes the traveler's frame from the Earth's frame Please read the article at https://www.scienceforums.net/applications/core/interface/file/attachment.php?id=18516 PDF: Twin paradox when Earth is the moving frame url removed or Word: https://www.academia.edu/39216040/Twin_paradox_when_Earth_is_the_moving_frame
3. ## Continuity and uncountability

Thanks Thanks I just want to know the name of this philosophy.
4. ## Continuity and uncountability

I agree with what you say. Is it a school of thought in mathematics? Are there many people who think like you ?
5. ## Continuity and uncountability

I know that this is the well accepted theory. But it would be great if the contrary could be proven. Yes, that is impossible. But one can always think otherwise.
6. ## Continuity and uncountability

I do not think that a line is a set of points just because points cannot be fill all holes. But this is another story. I agree that real numbers are countable.
7. ## Continuity and uncountability

I will change my theory to handle infinity.
8. ## Continuity and uncountability

No. Length is the number of member of a series. So, it is a natural number.
9. ## Continuity and uncountability

It seems that everyone thinks that natural numbers have finite values while the entire set in infinite. I'm OK with that. But in this case, the length of the set of all even numbers is finite, because it's a natural number. Actually, one cannot pass from a finite number, the length of a finite set, to infinite number, the length of a infinite set, which is the finite set when its length is stretched to infinity. Actually, if the price is 1111....., you can double it, 2*1111...=2222... Thanks for your help. I think within the set of natural number, the ordinal numbers are natural numbers.
10. ## Continuity and uncountability

I mean that because you say “i can be an arbitrarily large finite number.”, i and j cannot be infinitely big, for example, a number with infinitely many digits like 9517452…… or $$10^{\infty }$$. In this case, can we write the set of natural numbers as {1,2,3,…,n-1,n}, with n being a finite number with arbitrarily large value? If i had value 9517452…… , then the ratio i/1 would not have corresponding natural number.
11. ## Continuity and uncountability

I was restating your explanation above with my words. So, in the counting of $$\mathbb{Q}$$, the ratio i/j corresponds to the number n which is finite. For the ratio i/1, the number of count is: $n=\frac{i(i+1)}{2}$ So, if n is a finite number, i is also finite. Does this mean that the number i is not allowed to go to infinity? In this case, $$\mathbb{Q}$$ does not completely cover the plane $$\mathbb{N}\times \mathbb{N}$$ . Can we say that $$\mathbb{Q}$$ is countable only because i and j are not allowed to have infinite value?
12. ## Continuity and uncountability

Yes, you are right. So, the set of the rationals is countable because every i/j corresponds to a finite number n in the counting order. At nth step, we stop the count. Although the set itself is infinite, all counting numbers are finite. But for the power set of N, the set of all even numbers corresponds to the infinite binary sequence 1010101010...... To reach it we cannot stop counting because this sequence is not finite. Can we say that the power set of N is not countable because the counting numbers of infinite subsets are infinite? Yes. I agree that I have difficulty in seizing the exact sense of the discussion.
13. ## Continuity and uncountability

I'm very grateful to your help. Sorry for not replying you more.
14. ## Continuity and uncountability

Se cannot work with this "definition". Yes, your definition is correct. But is it used to prove that a set is countable? For example the rationals are countable. This is proved by counting along the diagonals of the plane N*N. The set {1/1,2/1,1/2...} is bijected to {1,2,3...n,...}, thus is countable. Is the definition used here?
15. ## Continuity and uncountability

Why X is infinite? If X={1,2} and Y ={1}, would X and Y fit this definition?
16. ## Continuity and uncountability

Can we have a definition of infinite set? Without a proper definition, work on infinite set does not have sense. Se cannot work with this "definition".
17. ## Continuity and uncountability

We agree on the definition of finite set. But what is the definition of infinite set? Like the set of all even numbers?
18. ## Continuity and uncountability

Like what we have to give a definition to what we are discussing in order to understand each other. Then, how can we discuss about infinite set? Can we say that "the infinitely many natural numbers" are all finite? That's not a precise definition. Can you give a precise definition in order to speak the same thing in our discussion? The context is about set of natural numbers. So, a set of different natural numbers that has an end.
19. ## Continuity and uncountability

A set that has an end. I don't know of it being defined at all. The disagreement between you two shows that infinity is really confusing. I agree. An infinite set "contains only natural numbers, and they are all finite, by definition. " It is infinite because it does not has end.
20. ## Continuity and uncountability

Infinity is not well defined. Is it an object with a value or is it a ever increasing number that is bigger than all? Does the set of natural number exist? If it exist, then, it contains infinite natural number. If infinite natural number does not exist, then the set of natural number does not exist because it does not contain infinity. I'm not sure if I understand you. What you mean with "That does not in any way match the definition you attempt to use to justify your claim there"?
21. ## Continuity and uncountability

Can you tell me how any of this works?
22. ## Continuity and uncountability

Graphic of set counting and infinite number When counting a set, we can plot a graphic that represents the members of the set on the plane (x, y) to observe visually the counting. Also, graphic of counting of infinite set helps us to understand infinite natural number. PDF Graphic of set counting and infinite number https://pengkuanonmaths.blogspot.com/2018/11/graphic-of-set-counting-and-infinite.html or Word https://www.academia.edu/37766761/Graphic_of_set_counting_and_infinite_number

Thanks.
24. ## Continuity and uncountability

Thanks. If there is not rule for an object, what will the rule that allows it to exist? An axiom? For example, the set of natural numbers is infinite. Nobody can write it down. So, if it is accepted to exist, it is because of an axiom, isn't it?
25. ## Continuity and uncountability

But does the set of even numbers exist? It is assumed to exist by an axiom which allows infinite set to exist. However, infinitely big natural number is not allowed to exist. If the set of even numbers exist, what is the number of its members? Aleph0 is not a natural number. We do not have bijection not because there is not, but because there is no axiom for infinitely big natural number to exist. This is an incoherence of set theory because a thing exists, the number of the members of infinite set, but this number does not exist in the same theory that creates this thing .This is the cause that no natural number corresponds to the set of even numbers.
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