« N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number»

“A concrete representation of a 4D Klein bottle has been desired by many but has never been presented. So, I decided to dive into the Klein bottle. Working with the Klein bottle was my first opportunity to practice with this system. To my surprise, the ease with which it allowed me to create 4D Klein bottles was remarkable. The 4D Klein bottles were generated smoothly without the slightest hitch. My video animations of the rotating 4D Klein bottle in 4D space, as well as the 3D slices ascending in the 4D space, were also computed effortlessly.”

Abstract: While a 3D complex number would be useful, it does not exist. Recently, I have constructed the N-complex number, which has demonstrated high efficiency in computations involving high-dimensional geometry. The N-complex number provides arithmetic operations and polar coordinates for N-dimensional spaces, akin to the classic complex number. In this paper, we will explain how these systems work and present studies on 4D Klein bottles and hyperspheres to illustrate the advantages of these systems.

The classic complex number system is a remarkable mathematical tool because it allows for the addition and rotation of vectors in two-dimensional space, following the same rules as real numbers for addition and multiplication. However, in three-dimensional space, it is impossible to manipulate vectors with similarly intuitive arithmetic operations because such a system does not currently exist. The development of a three-dimensional complex number system, analogous to the two-dimensional one, would represent a significant advancement in mathematics.

In 2022, I constructed a system of complex numbers for spaces with any number of dimensions, which I call the “N-complex number system.” Edgar Malinovsky used this system to create many beautiful 3D objects (see «Rendering of 3D Mandelbrot, Lambda and other sets using 3D complex number system»[4]). Figure 1 shows the 3D Mandelbrot set he created. Computing 3D fractal objects is very time-consuming; he would not have succeeded in this work without the 3-complex number system. His work demonstrates that the 3-complex number system significantly accelerates computations in 3D space.

I have worked on 4D Klein bottles by extending a 3D Klein bottle (see Figure 2) into 4D space. I rotated the 4D Klein bottles in 4D space and showcased the rotation in my video animation “Observing a 4D Klein Bottle in 4-Dimension” [5]. This work would have been impossible without the 4-complex number system. In addition to N-complex numbers, the new system provides a polar coordinate system for N-dimensional spaces, which was previously missing in mathematics.

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See « N-complex number, N-dimensional polar coordinate and 4D Klein bottle with 4-complex number» for more detail.

https://www.academia.edu/120524016/N_complex_number_N_dimensional_polar_coordinate_and_4D_Klein_bottle_with_4_complex_number

https://pengkuanonmaths.blogspot.com/2024/06/n-complex-number-n-dimensional-polar.html

Kuan Peng

Abstract: The walls of Tokamaks are heavily eroded by plasma which suggests that a mysterious force pushes the plasma to the wall. We have theoretically discovered this force and named it extra-force. This force appears only on single charges but not on current carrying wire, which is why the Lorentz force law does not contain it. This discovery not only brings new knowledge to electromagnetism, but also gives a solution against the erosion, which could improve the technology for controlled nuclear fusion. We have proposed an experiment to test this new force.

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A promising way to achieve controlled nuclear fusion is Tokamak which holds fusion fuel at around 100 million kelvins in a toroidal chamber. At this temperature the fusion fuel is in plasma state which is a mixture of positive ions and electrons. The plasma is confined with a powerful magnetic field which makes charged particles circling the confinement field lines. The coils of Tokamaks are so designed that the confinement field lines are parallel to the wall of the chamber and the Lorentz force is expected to retain the plasma not to hit the wall. However, the walls of running Tokamaks are heavily eroded by plasma as if charged particles do not completely obey the Lorentz force law. So, we suspect that on top of the Lorentz force a mysterious force is acted on the charged particles, which makes the particles escape the binding of the confinement field.

Previously, we have shown a flaw in the Lorentz force law, that it ignores the parallel-to-current force, see « From Coulomb’s force to magnetic force and experiments that show magnetic force parallel to current»^[1]. This flaw suggests that the mysterious force could also be  a force that the Lorentz force law ignores. In the following, we will derive the expression for the mysterious force by starting from the Coulomb’s law that expresses the electrostatic force that a charge q_b exerts on another charge q_a:

When q_a and q_b move, relativistic effects come into play and modify the Coulomb’s force given in (1). There are two such effects : the dynamic effect and changing distance effect. Changing distance effect means that when the distance between q_a and q_b changes, the force on q_a changes. Dynamic effect means that the intensity of the force is directly modified by the relative velocity.

See « Erosion protection of Tokamak and magnetic force on single charge » for more detail.

https://www.academia.edu/116178626/Erosion_protection_of_Tokamak_and_magnetic_force_on_single_charge

https://pengkuanem.blogspot.com/2024/03/erosion-protection-of-tokamak-and.html

Kuan Peng

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Letter to readers

Dear readers,

I have derived the extra-force because I thought that the magnetic force on a single charge could be different from the Lorentz force. Why did I have that intuition? We know that currents are composed of positive charges and moving electrons. So, the Lorentz force is surely the difference between the force on the positive charges and that on the moving electrons. Because the moving electrons have a velocity while the positive charges do not, it is surely the velocity that modifies the electrostatic force. The only way physical quantities could be modified in moving frame is through relativity. So, I have tried to modify electrostatic force with relativity and fortunately, was successful in deriving the 4 magnetic forces.

I have proposed to inject additional electrons into functioning Tokamak to protect its wall. This technique would not only make Tokamak work longer, but also allow higher temperature of the plasma because the latter would no longer touch the wall. This technique would increase the density of the plasma because the extra-force is a compressing force so that more plasma can be loaded. In consequence, the confinement time, temperature and density would all be improved at once, making the Lawson criterion and controlled nuclear fusion closer to reach.

The Maxwell’s theory is very successful for explaining the majority of electromagnetic phenomena, which makes us believe that it can correctly predict all electromagnetic phenomena. However, there are things it does not predict but really exist, for example, the extra-force. When these things happen, for example, the erosion of the wall of Tokamaks, we do not understand why they occur, let alone to fix their consequence if disastrous. So, it is important to correct the Maxwell’s theory when a flaw is found.

The missing of the extra-force is a flaw in the Maxwell’s theory and its discovery a breakthrough for electromagnetism. So, this article topples the Maxwell’s theory again and will be no doubt rejected by all respectable journals. In consequence, I will not submit it for publication but only post online. I’m sure that you, my online readers, would have the wise to recognize its value and would promote it either by sharing it with your colleagues or by performing the proposed experiments. If you succeed in doing the experiments, you will get honorable credit for your contribution.

Kuan Peng, 13 March 2024

See « Erosion protection of Tokamak and magnetic force on single charge » for more detail.

https://www.academia.edu/116178626/Erosion_protection_of_Tokamak_and_magnetic_force_on_single_charge

https://pengkuanem.blogspot.com/2024/03/erosion-protection-of-tokamak-and.html

Abstract: The Lorentz force law is fundamental for electromagnetism. However, it is known long ago that the Lorentz forces between two current elements do not respect the Newton’s third law. This seemingly harmless flaw had never been corrected. In physical sciences a discrepancy often hides in it new understanding or unexpected breakthrough. For solving this problem, we give a purely theoretical derivation of magnetic force which respects the Newton’s third law in the case of current elements and is identical to the Lorentz force in the case of coils. This new law reveals how electric force is transformed into magnetic force by velocity and is supported by experimental evidences that we will explain and compute with the new law.

1.Introduction

The Figure 1 shows a case where dF_a is perpendicular to dF_b , so, dF_a + dF_b ¹ 0. This problem was known for longtime. People justify that the Lorentz forces that two closed loop currents act on each other do satisfy the Newton’s third law. Nevertheless, breaking the Newton’s third law does not fit scientific standard, even for the Lorentz forces law which is fundamental.

We will try to solve this problem with a new magnetic force law that we have derived with pure theory. The new law is derived from the Coulomb’s law which defines the Coulomb’s force for fixed charges. For moving electrons, the Coulomb’s force undergoes relativistic effects and varies with velocity.

2.     Consequences

·       The relation mu₀ eps₀ c² = 1

Historically, the values of mu_0, eps₀ and the speed of light c were measured experimentally. It was James Clerk Maxwell who noticed that mu₀eps₀c² = 1 . So, it was an empirical law.  In our derivation this relation emerged naturally from both relativistic dynamic effect and changing distance effect. So, we have theoretically proven this relation and in consequence, the relation mu₀eps₀c² = 1 is now a theoretical law.

·       Biot–Savart law

The equation (58) is identical to the Biot–Savart law (59) but is derived with pure theory. So, the Biot–Savart law becomes a theoretical law too.

·       Lorentz force law

(61) is the Lorentz force that one dI_b exerts on dI_a . So, we have derived the Lorentz force law from the Coulomb’s law.

·       Magnetic force vs. Newton’s third law

The sum of the magnetic force (49) and its back force is zero. So, the magnetic force law (49) satisfies the Newton’s third law for current elements . Being an experimental law, the Lorentz force law does not describe a force that does not exist and thus, lacks this term. So, it cannot satisfy Newton’s third law. Thanks to the fully theoretical derivation, the magnetic force law (49) contains the missing last term and consequently, satisfies Newton’s third law.

3.     Experimental evidences

·       My experiments

The first experiment is «Continuous rotation of a circular coil experiment»[1]. The video of this experiment is: https://www.youtube.com/watch?v=9162Qw-wNow[2]. In this video we see a round coil that rotates in its plane. Because the coil is round the driving force must be parallel to the wire, that is, the driving force is parallel to the current. This force cannot be Lorentz force which is perpendicular to the current. A detailed technical explanation is in the paper «Showing tangential magnetic force by experiment»[3] .

I have also made a « Circular motor driven by tangential magnetic force »[4] . The video of this experiment is: https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ[5]. The technical details of this experiment is: « Detail of my circular motor using tangential force and the equivalence with homopolar motor » [6].

·       Experiment of wire fragmentation

In 1961, Jan Nasilowski in Poland has carried out an experiment which consisted of passing a huge current in a thin wire. The wire exploded into small pieces. The interesting thing is that the wires were not melted but teared apart by mechanical force.

4.     Conclusion

Because the new law gives the same prediction as the Lorentz force law for closed loop currents, it works for electromagnetism as the Lorentz force law. However, the component of magnetic force parallel to the current is new and shown to be rather significant. So, it could be used as the driving force for new devices.

Since the Biot–Savart law, the Lorentz force law and the relation mu₀ eps₀ c² = 1are derived with pure theory, the deep mechanism that transforms electric force into magnetic force is revealed to be the two relativistic effects, electromagnetism is much better understood.

For more detail of this study please read the complete paper here:

« From Coulomb’s force to magnetic force and experiments that show magnetic force parallel to current»

https://www.academia.edu/106863205/From_Coulombs_force_to_magnetic_force_and_experiments_that_show_magnetic_force_parallel_to_current Kuan Peng

[1] Kuan Peng, 2017, «Continuous rotation of a circular coil experiment», https://www.academia.edu/33604205/Continuous_rotation_of_a_circular_coil_experiment

[2] Kuan Peng, 2017, Video https://www.youtube.com/watch?v=9162Qw-wNow

[3] Kuan Peng, 2018, «Showing tangential magnetic force by experiment», https://www.academia.edu/36652163/Showing_tangential_magnetic_force_by_experiment

[4] Kuan Peng, 2014, « Circular motor driven by tangential magnetic force », https://www.academia.edu/6227926/Circular_motor_driven_by_tangential_magnetic_force

[5] Kuan Peng, 2014, Video https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ

[6] Kuan Peng, 2014, « Detail of my circular motor using tangential force and the equivalence with homopolar motor » , https://www.academia.edu/7879755/Detail_of_my_circular_motor_using_tangential_force_and_the_equivalence_with_homopolar_motor

Hilbert’s Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert’s Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.

Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.

Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.

Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.

In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert’s Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert’s Grand Hotel cannot accommodate additional guest in its infinitely many rooms.

For more detail of this study please read the complete paper here:

« Is Hilbert’s Grand Hotel a paradox? »

The apparent non‑gravitational acceleration the extra-solar-system ‘Oumuamua exhibits is puzzling. We find that when the position and velocity of the Sun is correctly set in computing the predicted orbit, ‘Oumuamua’s trajectory can be explained with gravity and we have reproduced the unexpected gap by computation. We also propose to search for new extra-solar-system high speed asteroids with SOHO to check our method with their trajectories.

1.     ‘Oumuamua’s acceleration

‘Oumuamua, formally designated 1I/2017 U1, is an interstellar object passing through the Solar System which was first detected by Robert Weryk using the Pan-STARRS telescope on 19 October 2017 [1][2][3]. It seems to exhibit non‑gravitational acceleration, making it go further than expected [3][4][5][6][7][8].

In the article « Our Solar System’s First Known Interstellar Object Gets Unexpected Speed Boost » June 27, 2018 [9], it was reported that “’Oumuamua had been boosted by 25,000 miles (40,000 kilometers) compared to where it would have been if only gravitational forces were affecting its motion”, see Figure 1 and Figure 2 which are screenshots of the animation in this article and show the predicted orbit and the unexpected gap of 40 000 kilometers between the last observation of ‘Oumuamua and the predicted spot.

Although “the Canada-France-Hawaii Telescope (CFHT) and, in the following days, the ESO Very Large Telescope (VLT) and the Gemini South Telescope, both8-meter-class facilities, found no sign of coma despite optimal seeing conditions”, the authors still “find outgassing to be the most physically plausible explanation” [5].

However, because of the lack of coma we think that the boost of ‘Oumuamua can still be attributed to some overlooked effects. For example, what if the Sun is moving in the solar system? What if, due to the motion, the Sun is not exactly at the focus of the predicted hyperbolic orbit? In these cases, the real trajectory of ‘Oumuamua will not fit the predicted one.

We know that the Sun is not exactly at the barycenter of the solar system and moves relative to it, as shown by Figure 3 [11] in which the Sun is the central point and the barycenter wanders around it. But in reality the solar system is an isolated system the center of which is its barycenter. The frame of the barycenter is inertial, in this frame the barycenter is fixed and the Sun wanders around it. So, the Sun is always at a distance from the barycenter and moves at nonzero velocity.

Pursuing this direction we propose this hypothesis: the unexpected gap could be the consequence of erroneous position and velocity of the Sun with which the predicted orbit was computed.

For checking this hypothesis we will compute the trajectory of ‘Oumuamua by adjusting the position and velocity of the Sun such as to reproduce approximately the gap of 40 000 km.

2.     Static and shifting orbits

The basic parameters of the predicted orbit for ‘Oumuamua are published by JPL / NASA in the page ‘Oumuamua (A/2017 U1) [12]. The eccentricity e, the semi-major axis a, the orbital elements [15] and the standard gravitational parameter of the Sun GM_☉ [13] are given in Table 1. The semi-latus rectum l and specific relative angular momentum h are computed in (2) and (3). The predicted orbit is a hyperbolic orbit which is expressed by equation (1) [14].

Just for the purpose of checking our hypothesis, we put the focus of the predicted orbit at the barycenter of the solar system. This orbit is static and we call it the static orbit. The orbit of ‘Oumuamua is a hyperbola the focus of which is the moving Sun. So this orbit shifts in the frame of the barycenter and is called shifting orbit.

…

3.     Search for high speed asteroids near the Sun

Beside of computing the trajectory of ‘Oumuamua, a better way to check our hypothesis is by experiment, that is, by observing new high speed asteroids and compare their trajectory with prediction. However, recorded asteroids and comets coming into the solar system from the outside are scarce. But I think that in reality such objects are not so rare, only that far from the Sun they are too faint to be detected. When they are near the Sun they become very bright and can be detected by Sun gazing satellites such as SOHO (Solar and Heliospheric Observatory).

Many such alien asteroids and comets are already recorded by SOHO and are dormant in great number in the archives of SOHO. Thanks to NASA Goddard’s YouTube Movie “Decades of Sun from ESA & NASA’s SOHO” [19], which is a video made with all the photos taken by LASCO/C3 between 1998/01/06 and 2020/10/23,  I have found several high speed asteroids and comets in it.

For example, the asteroid that was recorded from 2004/02/26 to 2004/02/28. I have computed its visual velocity which is the velocity of the dot on the image and got 160 km/s, see Figure 7. The comet recorded from 2015/02/18 to 2015/02/21 is measured at 182 km/s, see Figure 8. For comparison, the speed of ‘Oumuamua at perihelion is 87.71 km/s [8]. I used the diameter of the view field of LASCO/C3 which equals 30 radii of the Sun [20] and the time printed on the images to compute its visual velocity. As the actual trajectories make an angle with the plane of the image, their real velocity are forcefully bigger.  Only objects coming from the outside of the solar system can be as speedy.

I have made a clip of the asteroid and put it here: Super-fast alien asteroid (160 km/s), taken by SOHO in 2004, https://youtu.be/GTGuEKndNIc

The alien asteroids and comets in the archives of SOHO are interesting. We can count their number, mapping their direction and measure their record breaking speed and size. On the other hand, we can monitor in real time the appearance of new alien asteroids and comets, work out their orbits for observing them later.

In searching asteroids in the Movie “Decades of Sun from ESA & NASA’s SOHO” [19], the background stars and the streaks left by space particles are very dizzying which makes the researcher miss interesting asteroids and comets. So, I suggest that the background stars and streaks be removed for this research.

For those who are interested in seeing the asteroids and comets that I have found I have put in the appendices the links that point to the frames of the Movie “Decades of Sun from ESA & NASA’s SOHO” [19] where they appear. I have seen more comets because they are brighter and easy to see. I have also put the few visual velocities that I have computed, which show that some comets are very fast and could have come from the outside of the solar system.

4.     Discussion

The analysis above shows that the unexpected gap can be well explained by the gravity of the Sun provided that the position and velocity of the Sun be correctly set in computing the predicted orbit. What matters is that the position and velocity of ‘Oumuamua be given with respect to the Sun’s actual position and velocity. If ‘Oumuamua were located with respect to the barycenter of the solar system while the Sun is not there, the predicted orbit would be wrong.

We have shown that because the combined trajectory is not a hyperbola and has moving focus the non‑gravitational acceleration may be unnecessary to explain that “the observed orbital arc cannot be fit in its entirety by a trajectory governed solely by gravitational forces due to the Sun, the eight planets, the Moon, Pluto, the 16 biggest bodies in the asteroid main belt, and relativistic effects” [5]. This is corroborated by the lack of coma.

We have also discovered that even the smallest error on the position and velocity of the Sun is enough to create an unexpected gap. Indeed, the distance used to reproduce the unexpected gap is only 12% of the radius of the Sun. So, the position and velocity of the Sun and the data of ‘Oumuamua must be precisely set.

On the other hand, our hypothesis must be checked again and again before being validated. Indeed, other value was given to this gap, for example, in the article « THIS INTERSTELLAR ASTEROID IS ACCELERATING » [18], 100 000 km has been given to the gap rather than 40 000 km. Obviously, there are several different predicted trajectories for ‘Oumuamua with different results. The correct theoretical trajectory for ‘Oumuamua must give Gap = 0 and our method can help for finding it.

We have proposed to check our hypothesis using the SOHO satellite in real time to find new alien asteroids and comets, work out their orbits for observing them later. It will be interesting to search for alien asteroids and comets in the archives of SOHO, which I have done partly.

For more detail of this study please read the complete paper here:

« Trajectory of ‘Oumuamua and wandering Sun, alien asteroids and comets detected by SOHO »

https://www.academia.edu/100818112/Trajectory_of_Oumuamua_and_wandering_Sun_alien_asteroids_and_comets_detected_by_SOHO

The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive.

Roll, pitch and yaw are angles of orientation of an object in space and the conversion of these angles among different reference frames is not easy, as this example illustrates:

https://math.stackexchange.com/questions/1884215/how-to-calculate-relative-pitch-roll-and-yaw-given-absolutes

We will use 3D complex number to compute the roll, pitch and yaw of a telephon relative to a car. We label the car as object a and the telephon as object b. Their reference frames are labeled as frame A and B and these frames are orientated with respect to the ground whose frame is labeled as frame G. The base vectors of the frame A are a_x, a_y and a_z, that of the frame Bare b_x, b_y and b_z and that of the frame G are g_x, g_y and g_z. The vectors a_x, a_y and a_z and b_x, b_y and b_z are expressedwith g_x, g_y and g_z in equations (1) and (2) with M_aand M_b being the matrices of transformation.

The roll, pitch and yaw of the object b relative to the object a define the orientation of the frame B in the frame A. This orientation is defined by the angles of rotation q, j and y, see Figure 1. For computing these angles we express the vectors b_x, b_y and b_z with the vectors a_x, a_y and a_z in (3), (4) and (5). In (10), (11) and (12) the coefficients in the parentheses are dot products between a_x, a_y and a_z and b_x, b_y and b_z. The equations (10) , (11) and (12) are written in matrix form in (13) from which we extract the matrix of transformation M_ba shown in (14), see (3), (4) and (5).

From the 5 underscored coefficients in (14) we derive the angles q, j and y using (15), (16) and (17), knowing that  is always positive because j is between -p/2 and p/2, see Figure 1. Roll, pitch and yaw of an object are the rotation angles of the object around the x, y and z axis respectively, see Figure 1 and the page https://en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft).

So, the conversion formula between roll, pitch and yaw and the angles q, j and y is equation (18).

The main advancement given by our method using 3D complex number is to have related the roll, pitch and yaw of an object to the 3D complex number that represent the x axis of the object. This mathematical discovery makes the conversion of roll, pitch and yaw between arbitrary frames easier and more intuitive.

The angles of rotation q, j and y are computed with the 5 underscored coefficients of the matrix M_ba in (14) saving thus the computation time for the 4 other coefficients which are not needed. This will make the computation faster and the motion of moving images on screen smoother. So, our method using 3D complex number would be beneficial to applications such as video games or street view of digital map etc.

For more detail of this study please read the complete paper here:

«Determination of the relative roll, pitch and yaw between arbitrary objects using 3D complex number»

https://www.academia.edu/92242546/Determination_of_the_relative_roll_pitch_and_yaw_between_arbitrary_objects_using_3D_complex_number Kuan Peng

Cantor’s diagonal argument claims that ℝ is uncountable. When we see real numbers as points on the number line, we can put a name on each point and put the names into a list without contravening Cantor’s diagonal argument because we cannot create a diagonal from a list of names.

However, we do not need such a impossible list, but just to split R into two parts, S2 and S10. the members of S2 are real numbers expressed in binary, those of S10 in decimal. We create a list of real numbers by picking one member from S2 and one member from S10 alternately and forever. This list is a composite list whose members are in binary and decimal alternately. The diagonal of this list is a sequence of binary and decimal digits alternately and out-of-the-list-number cannot be constructed from it.

In fact, composite list can be created in splitting R into many subsets in numeral systems of different bases from which no out-of-the-list-number can be created and there is no real number excluded from the composite list. Because the composite list is constructed from the whole R and no real number is found outside, the composite list contains R.

If there is one list that contains R we can already conclude that R is countable. But the permutation of the subsets of R can create a huge number of different composite lists which all contain R. So, we conclude with confidence that R is countable. Then Cantor’s diagonal argument fails.

Cantor’s diagonal argument expresses real numbers only in one numeral system, which restricts the used list. If a binary list is shown not to contain R, this can be caused either by “list” or by “binary”. Because Cantor has focused only on “list” overlooking “binary”, this is the flaw that breaks Cantor’s diagonal argument which then does not prove ℝ uncountable.

For more detail of this study please read the complete paper here:

«Real numbers and points on the number line with regard to Cantor’s diagonal argument»

https://www.academia.edu/88279926/Real_numbers_and_points_on_the_number_line_with_regard_to_Cantors_diagonal_argument

An analysis of Cantor’s theory of uncountable sets: The logic of his proofs has some weaknesses. Cantor assumes for both his proofs that all real numbers (set R) are in a list (list L). Considering L as a set this assumption assumes R belongs to L. This makes the claim “a real number is constructed but is not in the list L” questionable. We propose a solution to this problem, an axiom for counting infinite sets and a solution to continuum hypothesis.

«Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets»

https://www.academia.edu/86410224/Examination_of_Cantors_proofs_for_uncountability_and_axiom_for_counting_infinite_sets

I do a detailed analysis of Cantor’s theory of uncountable sets. The logic of his proofs has some weaknesses. I propose an axiom and a solution to continuum hypothesis.

The main idea is:
Assumption of Cantor’s proofs: All real numbers (set R) are in a list (list L).

This assumption means R=L, considering L as a set. This makes the claim “a real number is created but is not in the list L” wrong. Indeed, if a number is outside L, it is outside R too. So, the statement “the created real number is not in the list L” means it is not in R and is not a real number, which is equivalent to claim that a real number is not real number. This is absurd but Cantor’s diagonal argument and nested intervals proof both claim that a real number is not in the list L and thus, is not a real number, which make them wrong.

On the other hand, Cantor’s both proofs search for contradiction. Can “this real number is not a real number” be the contradiction? No. The contradiction of the proofs is in the third step: sout “is not in the list”. By failing to create sout the second step collapses before the third step declares the contradiction.

See the paper
https://www.academia.edu/86410224/Examination_of_Cantors_proofs_for_uncountability_and_axiom_for_counting_infinite_sets

For Newtonian mechanics

The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics, which is explained below. Consider a big mass M which creates the gravitational acceleration a for a small mass m at the distance r from M, see figure 1 and equation (1) for gravitational acceleration a=GM/r2. For computing v the radial velocity of m in the gravitational field of M we integrate equation (1).

We compute for the case where m freefalls from infinitely far starting with zero velocity, see (7). With these conditions the radial velocity of m at the distance r₂ from M is computed in (8), v2 = 2GM/r2. Reversing (8), r₂ = 2GM/v2. The Schwarzschild radius of the event horizon of M is r_s such that the Schwarzschild factor equals infinity, see (10).

When v₂ equals the speed of light c, we apply v₂ = c into (9) and we obtain (11) where r₂ = r_s. So, r₂ equals r_s and the Schwarzschild radius is computed with Newtonian mechanics.

For relativity

Although the Schwarzschild radius r_s is a relativistic quantity, in the above it is derived completely with Newtonian mechanics, which is somewhat weird. What will be its value if we apply relativistic principle?

In the following derivation we will use the formula for relativistic transformation of acceleration which is derived in the paper « Relativistic kinematics » linked here: https://www.academia.edu/44582027/Relativistic_kinematics

The formula is the equation (18) of the paper.

Here, this formula is given by (12) in which the gravitational acceleration of m is a and the acceleration in space is a_r and the radial velocity is computed by integration. With the same conditions as (7), the constant of integration k equals 0, see (16). Then, using k = 0 in (15), v is expressed with r in (17).

In the case where the small mass m approaches M, the distance r approaches 0, the radial velocity of m approaches the speed of light c, see (18). So, v the radial velocity of m does not become bigger than the speed of light c for r > 0. The Schwarzschild radius r_s is the radius such that v = c. So, r_s = r = 0, see (19).

Conclusion

When the relativistic principles are applied correctly, the Schwarzschild radius r_s equals zero. The gravitational force on m approaches infinity near M. But the speed of m never reaches the speed of light c, which is true for any force however strong it is and for time of acceleration however long it is. The Schwarzschild radius r_s must obey relativistic principle and is shown to be zero.

That the Schwarzschild radius r_s equals zero means that the geometrical size of a black hole should be zero and thus, a black hole should not have an interior. I have reached this conclusion and explained it in the paper « Gravitational time dilation and black hole » in which I have also shown that point masses could not coalesce to form a black hole whose volume is zero and that observations support this conclusion. This paper is linked here:

https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

The equations and figure are in the paper below

https://www.academia.edu/84798805/Radius_of_a_black_hole_for_relativity_and_Newtonian_mechanics

For more detailed information, I invite you to read the two cited papers:

« Gravitational time dilation and black hole » https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

« Relativistic kinematics » https://www.academia.edu/44582027/Relativistic_kinematics