pengkuan

Thanks.

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?

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.

I cannot I cannot Sorry, This set is in fact that: {1,2,3...m-2,m-1,m} with m=2^n.

I cannot. No, there is not. I said "This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits" to answer your reply below. According to you, ...10101010 is not a natural number. So, ℕ does not contain numbers with infinite digits. In this case, only numbers with finite digits exist in ℕ. This is why I said "Suppose the number of members of N is n". If this means that there is a largest natural number, then this is derived from your reasoning. There can exist other way to accommodate sets with different size. I have proposed related set in my article. For example, {1,2,3...n} is related with {1,2,3...2^n}. When n becomes infinity, both sets are ℕ, but their sizes are different, while being both countable. "each of which has finitely many digits" means that they are all finite. n is meant to represent any finite number. Sorry, the letter N in my post that you reply meant ℕ. So, there is confusion.

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n. Then, the set of even numbers corresponds to 10…101010, with n digits.

Is {1,2,3,...} a finite set? This set is exhausted. I think that the set of all even natural numbers corresponds to the natural number ...10101010.

Building set and counting set A counting set is the set of natural numbers with which a countable set is put in bijection. Is the counting set for the power set of ℕ the set that builds it? Please read the article at Building set and counting set.pdf

I think you are replying my first post "Continuity and uncountability".

Yes, infinite sets are smaller than their power sets too. But I know that the power set of ℕ is countable.

When I said “The cardinality of finite sets does not have the same property than that of infinite sets” I meant the power set of finite set is always finite set. Finite sets have finite power sets and finite sets are smaller than their power sets. This should be a theorem because this can be demonstrated. So, there is no need of axiom to finite sets. But for ℕ it is different. ℕ is infinite and it cannot be proven that it can actually have a power set. This is why one has defined an axiom for the power set of ℕ. I take the example of the function y=1/(1-x) to illustrate that while for any x<1, y has a value, this does not imply that for x=1, y has a value. In the same way, the fact that finite sets can have power sets does not automatically imply that ℕ has a power set. ℕ is an infinite set, and infinity can break the mechanism of creating power set. This is why, the axiom of power set maybe wrong for ℕ and its power set.

The cardinality of finite sets does not have the same property than that of infinite sets. The cardinality of ℕ may be a upper limit. There exists limits for mathematical objects that are walls that one never crosses. Like number 1 for 1/(1-x).

Correct, the diagonal argument is NOT Cantor's theorem. Cantor's theorem uses the diagonal to prove its result. Take the subset that contains the indexes of all subsets of ℕ. This means that every subset has an index, thus, is a member of a list. Then, every subset contains or not its index, qualified by selfish or not-selfish. The diagonal bit of selfish subset’s binary string is 1, otherwise is 0. The diagonal is used here. It is correct that any set has strictly smaller cardinality than its powerset. Correct. I do not complain about the contradiction, but about the sense of the theorem, that is, does it prove uncountability? An axiom can be refuted, if the power set of ℕ is proven non existing. Yes The validity of this definition depends on the validity that uncountability exists. Cantor's theorem proves uncountability using uncountability. This is not allowed.

A: Cardinality of the power set is bigger than that of ℕ . The power set is uncountable. B: Uncountability exists. A is true: "the power set is uncountable", only if B is true, that is, uncountability exists. B is true: uncountability exists, if A is true: "Cardinality of the power set is bigger than that of ℕ = the power set is uncountable". But "Cardinality of the power set is bigger than that of ℕ" does not necessarily mean "the power set is uncountable"

If A is Proven by B, B should not be consequence of A. But apparently, that the power set is uncountable is consequence of the existence of uncountability. But uncountability is proved by the existence of uncountable the power set. Before Cantor, uncountability did not exist. Also, I have shown in Analyze of the proof of Cantor's theorem.pdf that, not using Cantor's theorem, the ℕ is smaller than the power set. So, before using Cantor's theorem, this is already a proven fact. Below is a short summary of the first error I have found in Cantor’s theorem. Cantor uses a proof by contradiction, which is the following. The proposition to be proved P: A list cannot contain all subsets of ℕ. 1. Assume that P is false. That is: a list L contains all subsets of ℕ. 2. A subset K is created. It is shown that K is not in L, which shows P is true. 3. P cannot be true and false at the same time, So, P is true. See section “Recall of the proof” of « Analysis of the proof of Cantor's theorem » The scheme of logic is: 1. P is assumed false 2. P is shown true. 3. Then P must be true. This logic holds if P is correctly stated. What if the statement of P is incorrect? In this case, will the proof fall apart? The error I have found is that the used list in P is limited in length by the creation of K. In section “Case of infinite set” of « Analysis of the proof of Cantor's theorem » I have shown that for a subset to be selfish or non selfish, the binary string of the subset must have the diagonal bit which equals 1 or 0. As the list L contains only subsets that are selfish or non selfish, the binary list L’ contains the diagonal. The length of the diagonal equals the length of the list L’ and the length of ℕ which is the width of the list L’. So, the length of L equals the length of ℕ. In this case, the proposition “A list cannot contain all subsets of ℕ” is not correct, because in the derivation the use of the property “selfish or non selfish” imposes the length of the list L to equal the length of ℕ. So, the length of the list in the proposition P is not free of constraint, but is imposed by another element in the same context. One can argue that because all subsets of ℕ cannot be put in the list L with the length of ℕ, subsets of ℕ are uncountable. But is ℚ countable? ℚ is created by one ℕ in x-axis and another in y-axis. In this case, ℚ is a list whose length is a free variable not limited by ℕ of its axis. Must the length of ℚ equal the length of its axis?

The proof is correct. But the premise is apparently wrong. The theorem states: the power set of ℕ etc. Does the power set of ℕ exist at all? If it exists, its cardinality would be aleph 1. But the existence of cardinality aleph 1 is proved by the uncountability of the power set of ℕ.

Analysis of the proof of Cantor's theorem Cantor's theorem states that the power set of ℕ is uncountable. This article carefully analyzes this proof to clarify its logical reasoning Please read the article at Analyze of the proof of Cantor's theorem.pdf

Abstract for the above post Theoretical explanation of longitudinal magnetic force and its practical application in high field magnet. Although longitudinal force is not explained in classical theory, its action has been demonstrated by several experiments long time ago. For example Nasilowski effect. But why is it not recognized in theory? The reason is that it shows no significant effect on practical devices, so no physicist is interested in exploring these experiments. But I have found a huge effect of longitudinal force in high field resistive magnets which could improve their performance.

Longitudinal magnetic force and high field magnet Theoretical explanation of longitudinal magnetic force and its practical application in high field magnet. Please read the article below Longitudinal magnetic force.pdf

Showing tangential magnetic force by experiment Theoretical explanation of tangential magnetic force and the experiment of rotating coil. Tangential magnetic force is tangent to the current on which it acts. For the classical theory this force does not exist. However, my experiment « Continuous rotation of a circular coil experiment » showed that a force tangent to the current must be there. If tangential magnetic force exists, why was it not detected in almost 200 years? Please read the article Showing tangential magnetic force by experiment.pdf

Plasma under Coulomb magnetic force Nuclear fusion reactors use strong magnetic field to confine plasma in reaction chambers. The magnetic field is so designed that plasma should follow field lines which do not encounter the chambers’ wall. But it seems that a mysterious force pushes plasma off its track. For explaining this force, Coulomb magnetic force law for plasma is derived. Please read the article at Plasma under Coulomb magnetic force Plasma under Coulomb magnetic force.pdf

Coulomb magnetic force The relativistic length contraction effect and changing distance effect produce 2 different magnetic forces. Together they form complete magnetic force. I have derived 2 magnetic forces with Coulomb’s law and charges’ velocity. The first force dFlc is derived in «Length-contraction magnetic-force between arbitrary currents». The second force dFcd is derived in «Changing distance effect». dFlc and dFcd are added together to give the expression for complete magnetic force dFcm. Please read the attachment Coulomb magnetic force.pdf Changing distance effect.pdf

Yes, any new theory originates from new approach. As I stated, electrons goes under relativistic length contraction and make the wire seems to have excess of charge, a current carrying wire will create an electric field around. The fact that such electric field is not predicted by classic theory does not mean that this field does not exist. Simply in 200 years, no one has tested this, at least no experiment subsists in textbook. It is a good idea to do an experiment to show this field. I will do it in the future. My last experiment showed that tangential magnetic force exists, which is in contradiction with classic electromagnetic theory. So, it would not be totally surprising if this electric field shows up,

In my article, SR is only mechanical, that is, length contraction and time dilation. Your are saying about relativistic electromagnetism which uses Lorentz force law in high speed. But this is not in my consideration. What I'm doing is to find magnetic effect from length contraction and Coulomb's law. So, a priori, there is not magnetic field in space and no relativistic electromagnetism at all. Electromagnetism comes after relativistic effect, not before. The two parallel moving charges are not in the domain of my theory, but in the domain of classic relativistic electromagnetism. What I'm doing is to compute a force on an element of current, which is acted by another element of current. This force integrated over a closed circuit equals that computed from Lorentz force law. So, this force is surely a magnetic force while magnetic field was not in the condition, which comes after the force on elements of current. Magnetic field is computed from the force, not the force computed from magnetic field. This is another approach of magnetism.

It's OK that you are not convinced. For the two charges, I do not see why the force between o and o moving i parallel decreases according to SR, since their distance is perpendicular to their velocity.