pengkuan

Just about GR and rotating disc which are the point of this discussion. GR is generally thought as specific to gravitational field and in an electric circuit there is not this GR. I cited GR just to illustrate non-euclidean geometry. I apology for confusing the subject by using the word GR. It is in fact not GR, but non-euclidean geometry or metric of a rotating disc that solves this problem. The system of a rotating disc is indeed a non-euclidean space and a rotating circle in this geometry does not has the same length than the same circle in a stationary state. See https://en.wikipedia.org/wiki/Ehrenfest_paradox. I use non-euclidean geometry to explain that the same number of charged particles in a same lengthed circle can have different density because the lengths are evaluated in 2 different spaces using 2 differents metrics, one euclidean the other non-euclidean. I agree that this explanation does not convince everyone.

Thanks for discussing this point. This is indeed a subtil relativistic phenomenon. How does the density of charge change while their number does not increase. One explanation could be that the length of the moving system of reference contract, not only locally, but globally. Let us see a circular circuit in which the electron move at the same speed. At any point, the length is contracted in the same ratio. So, the total length of the circle made by the moving electrons is: [latex]l_{e}=l_{0}/\gamma[/latex] where [latex]l_{0}[/latex] is the length of the circuit and [latex]l_{e}[/latex] the length of the moving circle of electrons. The contraction of the total length makes their density bigger in the fixed system. But how can the same circle have 2 different length? I think General relativity can explain this. In fact, Einstein used the system of a rotating disc to study General relativity. He solved the problem of the incompatibility of length of the 2 systems by stating that the space is curved in the moving system. Around a star, the circular trajectory of a planet has a length in the gravitational field which is different from that in a flat space. In the case of current, the space of the moving electrons is curved and its apparent length is contracted everywhere in the fixed system. Then, the density of the electrons is higher in the fixed system. The moving system is a circle of a disc's system whose curvature is proportional to the speed of the electrons. One may think that curvature of space is farfetched to explain the length of electrons' system. But this is one of possible explanation which is accepted for gravitational field.

Continuous rotation of a circular coil experiment There is a long standing debate about whether tangential magnetic force exists. In «Tangential magnetic force experiment with circular coil» I discussed this force and presented an experiment that showed the action of this force. But, as the rotation of the coil in that experiment was limited to a small angle, it does not show that tangential force exists all over the coil. So, I have carried out the present experiment that shows continuous rotation of the coil to make clear that tangential force has the same value around the coil Please read this article Continuous rotation of a circular coil experiment.pdf

Tangential magnetic force experiment with circular coil If magnetic force is to respect Newton’s third law, there should be a recoil force on the vertical current which is Ft. This force is tangent to the current I1 and called tangential magnetic force. Some physicists claim that tangential magnetic force exists, this claim is supported by some experiments such as the rail gun recoil force shown by Peter Graneau and Ampère's hairpin experiment, see Lars Johansson’s paper. But these experiments did not convince the main stream physicists and tangential magnetic force is rejected. I have carried out an experiment to show tangential magnetic force acting on a circular coil. Please read the attached document Tangential magnetic force with round coil.pdf video url deleted

Thanks. I know that Feynman has explained this principle for infinitely long current.

Thank you. My object is to derive the basic laws of magnetism from Coulomb's law and relativity, That is, Lorentz force and Faraday's law.

Length-contraction-magnetic-force between arbitrary currents In ≪Relativistic length contraction and magnetic force≫ I have explained the mechanism of creation of magnetic force from Coulomb force and relativistic length contraction. For facilitating the understanding of this mechanism I used parallel current elements because the lengths are contracted in the direction of the currents. But real currents are rarely parallel, for example, dIa and dIb of the two circuits in Figure 1. For correctly applying length contraction on currents in any direction, we will consider conductor wires in their volume and apply length contraction on volume elements of the wires. Please read the article here Length-contraction-magnetic-force between arbitrary currents.pdf

I'm sorry. I have uploaded the file here.

length contraction and magnetic force.pdfMagnetism is intimately related to special relativity. Maxwell's equations are invariant under a Lorentz transformation; the electromagnetic wave equation gives the speed of light c. Many have explained magnetic force as a consequence of relativistic length contraction, for example Richard Feynman in page 13-8 of his ≪The Feynman Lectures on Physics, Volume II≫ and Steve Adams in page 266 in his ≪Relativity: An Introduction to Spacetime Physic≫. If magnetic force is really created by relativistic length contraction, we should be able to derive the expression for magnetic force from the length contraction formula. And indeed we can, as I will show below. Please read the article at PDF Relativistic length contraction and magnetic force URL deleted

Thanks for asking this question that makes things clearer. Yes, they have a point in common. This means that line includes points. But line is not made of points, which was not explained by me previously. Let me try with an example. Take a square of sides s and 2 ribbons of width s. When 2 ribbons intersect, the intersection is a square. So, I will say that a ribbon is "made" of squares because when the squares are put one next to another, we will have a ribbon. Let us shrink the value of s to zero. The square becomes a point that has neither width nor length. The ribbon becomes a line with no width. In this case, the intersection of 2 lines has neither width nor length, it is a point. But if we put the point one next to another, we will not get a line, because zero length added to zero length gives zero length. This is what I mean by "line is not made of points." But line does include points.

Thank you for your reply. The main idea I want to show in my paper is the difference of characteristic between a one dimensional line and a 0 dimensional point. As the point's length is zero, adding points in the dimension of the line is equivalent to adding 0's, which gives zero length at the end. So, a line is not made by points and Cantor's effort to make line with uncountable points is not valid. But in a discussion in an other forum, I learned that using continuity to qualify line is not appropriate for explaining my idea. In fact, continuity is a property of string of points and I cannot use it to show the difference of characteristic between line and point. I will try using another method another day.

This is surely the best thing to do. I agree.

You are right to point out that the real line is not an accurate model of the real world and non-Euclidean geometry is a good example. One of the role of mathematics is to provide accurate model of the real world to physics. If there is a physical phenomenon that does not have a accurate mathematical model for representation and prediction, then someone will surely invent one. Like non-Euclidean geometry which represents spacetime for Einstein, a type of continuity different from Euclidean space and real line must be invented to represent continuous line such as electrical circuit where electrons do not jump from point to point, or trajectory of the Earth that does not jump in space. Because real line is discrete, a moving point must jump in order to move. Other examples of geometrical continuity such as iron chain made of links or pearl necklace exist These are continuous lines made of discrete objects and surely possess interesting properties. The interesting thing in mathematics is not to say whether a idea conforms today's mathematical theories or not, but if the idea can be the base of new things that will built upon the new idea. Construction of new is the big thing. New things are always nonsensical for old mathematics, as non-Euclidean geometry for Euclidean geometry or root -1 for real number theory. Great mathematicians are those who construct new mathematical theory, those who check mathematical error are good professors. What I do here is to propose new ideas. But I'm unable to construct mathematical building.

By holes in the rational lineup I mean the holes left by irrational numbers that are not in the rational lineup. Thanks wtf for your long reply. By holes in the line of rational number, I mean the points that irrational numbers occupy. In fact, I do not think that uncountable set is continuous, rather the contrary, uncountable set is not continuous, even the real line. However, I write in the introduction the idea of continuity of the mainstream idea because I cannot make people agree with my point from the beginning. I say what they agree with, then I deduce the point of none line and none continuity. I have said about Cantor’s ternary set in the end of my paper. The rest of my paper explains why real line is not a line and why it is not continuous. Yes that the standard construction of the real numbers does not need infinitesimals. But the real line is not continuous. This is why I create the definition of continuity, which needs infinitesimal length because real numbers have zero length and zero length does not allow continuity. If you draw a line that is the path of a point through space, then between points there is not continuity. See the electron that passes from one point to the following, it will not be able to forego the point in between. I do not say "continuity" of a point set, but continuity of a line. In the section 3 “So, I propose the following definition of continuity: A line is continuous between 2 points C and D if the space between them is zero. Equivalently, the line is continuous between C and D if a moving point can go from C to D without crossing other point else than C and D. If all points of a line satisfy this condition, then the line is everywhere continuous.” Line is not made by point, is not a set, but a geometric form. Line can be continuous , not set. I mean that we cannot fix all the digits of the limit because its last digit cannot be found. Then the limit is not a real number. This is why this limit does not exist and Cantor's claim that there is forcefully a limit is wrong. Cantor said Real numbers are a continuum, I take this as a claim that ℝ is continuous. I'm not discussing topology, I do not know it. But if uncountability is not true, then cardinality is not true either. In fact, I define a line not as a string of real numbers, so the points of the line cannot be real numbers. This is why I call the "points" blocks and which have infinitesimal length. Please think the real line as a line that is fully filled, not only by real numbers that are discrete points. uncountability is a property of set. But continuity is that of geometric line, which is not built by points, that is , by set. I think the notion of continuity that I have explained is not in the present mathematics.

Continuity and uncountability Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis. The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together. Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean? Please read the article at PDF Continuity and uncountability http://pengkuanonmaths.blogspot.com/2016/09/continuity-and-uncountability.html or Word https://www.academia.edu/28750869/Continuity_and_uncountability Continuity and uncountability Peng Kuan 彭宽 27 September 2016 Abstract: Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis. What is uncountability for? The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together. Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean? Cauchy’s continuity I haven’t found existing definition for continuity of line but definitions for continuous function instead. For example, using a Cauchy’s sequence s=(x_i | x_i∈R)_(i∈N) which converges to a point a, the continuity of a function f (x) at point a is defined as follow: lim┬(i→∞)⁡〖x_i 〗=a⇒ lim┬(i→∞)⁡f(x_i )=f(a) (1) The real line is the function f=0, satisfies this definition at any real numbers and is continuous. I call this definition Cauchy’s continuity. However, if a and xi are rational numbers, the converging sequence will be entirely in ℚ and Cauchy’s continuity will allow the set of rational numbers to be continuous, which is wrong. So, Cauchy’s continuity is inadequate to define continuity of line. Geometric continuity Figure 1 Line is a geometric form that represents the form of real objects, for example conductive wire, water pipe, trajectory of planets etc. To illustrate the continuity of line, imagine the lines in Figure 1 as a conductive wire interrupted between points A and B. When electric current flows in the wire and the interruption, electrons move in the conductive medium of the wire and make an electric arc through the air in the interruption. To cross the interruption an electron must quit the conductive medium from point A, pass through the air and enter the point B. Following this image, continuous line is a mathematical medium in the form of line in which a point can move without quitting. An interruption is a location where a moving point must quit the medium. So, I propose the following definition of continuity: A line is continuous between 2 points C and D if the space between them is zero. Equivalently, the line is continuous between C and D if a moving point can go from C to D without crossing other point else than C and D. If all points of a line satisfy this condition, then the line is everywhere continuous. C and D are said to be in contact and adjacent to each other. In the following, this kind of continuity will be referred to as geometric continuity. Real line Is the real line geometrically continuous? No interruption can be found on the real line, but the condition of geometric continuity is not satisfied. Take 2 different real numbers a and b and bring them close to each other, no matter how close they are, they are always separated by infinitely many other numbers. If an imaginary electron goes from a to b, it must cross many other points else than a and b. So, the real line is interrupted between a and b but not geometrically continuous. Also, being not in contact with other point, a is an isolated point. As a can be any real numbers, all real numbers are isolated and the set ℝ is discrete. So, ℝ is not a continuum. Constructing continuous line Figure 2 Why are real numbers discrete? Let us see Figure 2. The points on the right are in contact to each other and they are continuous. The distance between the centers of adjacent points is denoted by d and the width of points by w. These points are continuous because w=d. On the left, the distance between the points is still d but the width of points is smaller, w<d, this makes them discrete. However, we can shrink the distance d on the left to make the points continuous again. If the points were real numbers, the width of points equals zero and, however small the value of d is, the points are always separated by a distance because d>0. Therefore, that the width of points is zero is the reason that makes real numbers discrete. This also proves that uncountability is unrelated to continuity. Indeed, real numbers are uncountable and discrete at the same time. On the other hand, if one puts a real number s in contact with another number r, they will occupy the same point because their widths are zero. If t is put in contact with s, the 3 numbers r, s and t will occupy the point of r. We can repeat this operation uncountably many times, we will obtain only one point, not a line. So, uncountably many points of zero width do not make continuous line. Figure 3 Figure 4 So, to construct a geometrically continuous line the constructing points must have nonzero width, that is, w>0. What would be the value of w? Let us deconstruct the continuous line in the interval [0,1] by splitting, as shown in Figure 3 and Figure 4. The first splitting point is ½, then the resulted 2 segments are split at ¼ and ¾. And then, the 4 resulted segments are split at ⅛, ⅜, ⅝ and ⅞. The spitting goes forever and we obtain an infinite sequence of splitting points ssplit=(aiℝ )iℕ and an infinite sequence of segments. The segments are in contact with one another, securing continuity. Their length equals the infinitesimal number ε=1/2^∞ . These segments are the constructing blocks of the original line, each one starts at its splitting point aiℕ and has the length . Remark: The construction of geometrically continuous line proves that the controversial infinitesimal number  really exist, otherwise, continuity cannot arise. General model Figure 5 For a general line in space such as the one shown in Figure 5, a constructing segment is determined by 6 quantities: 3 coordinates for starting position, 2 angles for direction and  for length. This segment, S in Figure 5, will be referred to as infinitesimal vector-segment and is the constructing blocks for general line. Real numbers are discrete points that are 0-dimensional objects. In the contrary, infinitesimal vector-segment has nonzero length and is a one-dimensional object. So, we have the following property: One-dimensional geometrically continuous line is constructed only with one-dimensional objects. Consequently, 0-dimensional points cannot construct one-dimensional line, even they are uncountably many. In general, continuous objects in higher dimension are not constructed with objects of lower dimension. For example, 2-dimensional surfaces are constructed with infinitesimal surface2 and n-dimensional volumes with infinitesimal n-volumen. Uncountability How did Georg Cantor link uncountability to continuity? In fact, he constructed the continuum ℝ in two steps: 1) ℝ is uncountable; 2) Uncountability of ℝ creates continuity for the real line. Figure 6 He concentrated himself on proving that ℝ is uncountable. The first proof he gave was based on nested intervals [a0, b0], [an, bn], as shown in Figure 6. Because anbn when n, Georg Cantor claims that the limit of an and bn is a number not included in the lists a0a and b0b, thus real numbers are uncountable. However, does the limit of an and bn really exist? A limit is a real number which must be fully determined, that is, all the digits from 1st to th are fixed, for example . When n increases, the first m digits of an and bn get fixed and make a number that seems to converge. The first m digits of the limit may equal this number, but the limit’s last digits, from m+1st to th, will never be determined. In fact, when n increases, an and bn both vary and the points within the interval [an, bn] are all undetermined. So, the limit that Georg Cantor claims cannot exist and this proof is invalid. In addition to this flaw which is explained in «On Cantor's first proof of uncountability», Georg Cantor’s later proofs, the power-set argument and the diagonal argument, contain also flaws, which are explained in «On the uncountability of the power set of ℕ» and «Hidden assumption of the diagonal argument». So, all 3 proofs that Georg Cantor provided fail and uncountability possibly does not exist. About the second step Georg Cantor did nothing but simply claim that ℝ is a continuum; probably he assumed that uncountability really created continuity. But it is shown above that uncountability is not related to continuity. So, uncountability has lost its utility and becomes useless except for itself. Continuum hypothesis The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers which is a discrete set and the real numbers which is a continuum. The idea behind this hypothesis is that there cannot be set that is discrete and continuous at the same time. Georg Cantor tried hard to find such set; the Cantor’s ternary set is probably one of his attempts. However, it is shown that uncountability is not proven, then the cardinality of real numbers is questionable. Anyway, ℝ is not a continuum and the continuum hypothesis makes no longer sense.

Rank of infinite numbers can be arranged because we can count them. Infinity cannot be count, so, we are dealing with something that is not dealable. This is why either CH is true or false does not matter. Infinity is a philosophical idea, which is qualitative. Mathematics deal with quantitative things, that infinity is not. However, I have learned a lot about mathematics. From my first wrong definition of discreteness that makes irrationals discrete to the present better definition using void space. It is a great leap.

Thanks for joining. I think they do not discuss the truth of CH because they they think my proof is not valid not because they are not interested in this topic. Continuum hypothesis is famous because it is the first in the list of Hilbert, but how will it influence mathematics is not clear. One is interested in this topic like a bounty hunter. Can someone explain how mathematics will benefit from continuum hypothesis?

I'm sorry to disappoint you like this. I have not sufficient mathematical knowledge to write a proof at the standard of mathematics and I cannot swallow the necessary quantity of mathematical knowledge in one day while others learn it in several years. I will let this paper as it is now and move on to something else.

I have posted my detailed explanation before your 'rough guide' because the administrator have given me a note that warned me not to force people to open my site or pdf. So, I have to obey quickly. Since then , I have seen your opinion on compactness, boundary, adjacent or neighborhood. I have seen that the continuity that I used to qualify real is equivalent to compactness. So, I have added this aspect to my paper. Also, you precise that neighborhood is a notion of measure theory that has not meaning here. So, I have corrected my paper to avoid this notion. In fact, neighborhood is not necessary to qualify compactness. Instead, I have qualified discreteness by the two void spaces around a elements and the contact in real line with non-presence of void space around a element. So, I have avoid the notion of adjacent, neighborhood, contact. Nevertheless discreteness and continuity can be qualified by these properties. Below is the new version of my paper. By correcting according to your objections and other's, I'm zeroing in a definition of discreteness and continuity more acceptable. 1. Discreteness of sets Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named (0, 1). Then we put the number ½ in its middle to make a set with three elements {0, ½, 1}. The new point ½ cut the void space (0, 1) into 2 void spaces, (0, ½) and (½, 1). The set {0, ½, 1} is discrete because ½ is separated from the points 0 and 1by the 2 void spaces it created. We keep adding new numbers to the above set and obtain: {0, 1/4, ½, 1},{0, 1/4, ½, 3/4, 1}….Each new number splits a void space into two void spaces which keep the number isolated. For a set with n elements, there are n-1 void spaces. Each number is surrounded by the 2 void spaces it creates. Like water surrounding isolated islands, the void spaces surrounding each element make the elements isolated. So, the resulting set is entirely discrete. Suppose we have an discrete set with i ordered elements: { x0, x1 , x2 ,… , xi}. The next set having i+1 elements is created by putting an element x in the void space (xk, xk+1) such that xk < x <x k+1. By repeating this process indefinitely, the resulting set will have infinitely many elements. As the elements are added one by one without end, the cardinality of this set is À0. This set is also discrete because there is a void space between any 2 elements. In fact, discrete sets do not need to be constructed this way. It is enough that any element is surrounded by void spaces on both sides and the set will be discrete, no matter how the elements are arranged. For example, any rational number is surrounded by void spaces. In spite of the ever shrinking interval between 2 rational numbers that makes them infinitely close, the set of rational numbers is discrete. 2. Continuity of sets In the contrary, compact set is continuous, for example the real numbers. The elements of such set touch one another, that is, between any 2 points in the real line there is no void space in which an external point can be inserted. It is equivalent to say that except of boundary points, both side of a point x are immediately filled with other elements of the set, no void space exists there. As x can be any point in the real line, real numbers are in contact with one another and the set of real numbers is entirely continuous. This property defines the continuity of a set. 3. Collectively exhaustive and mutually exclusive events When tossing a coin, all possible outcomes are heads or tails. The values heads or tails are said to be collectively exhaustive, that is, there is no other possibility. Also, when heads occurs, tails can't occur and vice versa. These two values are said to be mutually exclusive, that is, the outcome is either heads or tails, no mixed value is allowed, for example half heads and half tails. If the elements of a set have void spaces on both sides, the set is discrete. If the interior elements of a set have no void spaces surrounding them, the set is continuous. The presence or not of the void spaces makes discreteness and continuity of a set the 2 outcomes of a collectively exhaustive and mutually exclusive game, no other possibility exists. 4. Continuum hypothesis The cardinality of an infinite discrete set is À0, like the rational numbers. The cardinality of a continuous set is |ℝ|, like the real numbers. As shown above, a set must be exhaustively and exclusively discrete or continuous. So, its cardinality must be À0 or |ℝ| but not strictly between À0 and |ℝ|. In consequence, the continuum hypothesis is true. I think void space alone have more sense than space void space that contains nothing but have nonetheless a length. I will read connectedness, compactness, completeness and coverings in relation to set theory. Let me more time, as I have to answer your messages also. I do not use neighberhoods any more. Can Countability be a proof of the discreteness? Rational are dense but do not contain their limits that are holes.

Can we create all possible discrete set? Can all discrete set be created? If we can prove this two propositions, then there is no difference on how discreteness is defined. But no answer exists. What I can say only is all so created set are discrete. Cantor's set contains continuum because it is constructed from a continuum by leaving 2 continuums in 3. So it is not discrete. The more advanced in searching in discreteness, the less I'm sure what is discreteness. I have the feeling that the limit of a discrete set is a continuum. Yes, I have not the correct word to name a void space that contains nothing but have nonetheless a length. So, you say that there is only one empty set and (1, 1/2 ) and (1/2, 1) are in fact the same empty set. Agree. How then we name the empty space (1, 1/2 ) and (1/2, 1) since they are not sets? Thanks. I have read your text but I can follow.

1. Discreteness of set Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named empty interval, here it is ]0, 1[. Then we put the number 1/2 in the middle to make a set with three elements {0, 1/2, 1}. The new point cut the interval ]0, 1[ into 2 empty intervals, ]0, 1/2[ and ]1/2 , 1[, which surround the point 1/2 . The set {0, 1/2 , 1} is discrete because 1/2 is separated by the 2 empty intervals it created from touching the points 0 and 1. We keep adding new numbers to the above set and obtain: {0, 1/4 ,1/2 , 1},{0, 1/4 ,1/2 ,3/4 , 1}….Each new number splits an interval into two intervals which keep the number isolated. For a set with n elements, there are n-1 intervals. Each number is surrounded by the 2 empty intervals it creates. Like water surrounding isolated islands, the empty intervals surrounding each element make them isolated. So, the resulting set is entirely discrete. Suppose we have an discrete set with i ordered elements: { x0, x1 , x2 ,… , xi}. The next set having i+1 elements is created by putting an element x in the interval ]xk, xk+1[ such that xk < x <x k+1. By repeating this process indefinitely, the resulting set will have infinitely many elements. As the elements are added one by one without end, the cardinality of this set is À0. This set is also discrete because there is an empty interval between any 2 elements. This property defines the discreteness of sets. Natural numbers and rational numbers are infinite discrete sets.

Agree. It is learning course to expose my idea and receive critiques. Before you explain me that discreteness is not as simple, I just explain intuitively. This is why I make confusion. But the confusion will fade with your help and from others, leaving the good substance clear. I must change my terminology. Using "may be " in place of "be"

Yes. I agree. I try to figure out another definition.

Yes. But real numbers are continuous, because between any two reals are only reals. The question is, how irrationals are uncountable? Isn't it?

Yes. I'm writing it. Its a very big table.