PengKuan’s work On Mathematics

 

Is Hilbert's Grand Hotel a paradox?

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

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Letter about point and real number

I will give my opinion on your reply paragraph by paragraph about my paper

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

 

1.       When we use the word "restrict" in a mathematical argument, we mean that we start with a set and then take a proper subset (i.e. a subset that leaves out part of the original set). 

Yes, By "restrict" I mean that real number can be expressed in many bases, such as bases like binary or decimal, base 30, base n etc. Cantor restricts to only base 2 or 10. So, in cases where lists with alternate binary and decimal expression, out-of-the-list-number cannot be generated, Cantor’s diagonal argument does not apply.

 

2.       Given any number base b, one can prove that every real number (say in the unit interval) can be uniquely expressed in this base as an infinite series, where each term in the series is a fraction in which the numerator is a digit in that base (0 through b-1), and the denominator is b to some positive-integer-power. (When I say "uniquely" I assume we disallow any sequence of numerators that end in all (b-1)'s, as I explained in my previous email.)

 

The composite list written with multiple numeral systems does not generate out-of-the-list-number. But list written with only one numeral system is a particular case that we have to deal with because you said “one can prove that every real number (say in the unit interval) can be uniquely expressed in this base”. So let us drop multiple numeral systems and see mono-numeral system.

 

Let Cantor create out-of-the-list-number with n real numbers.

The out-of-the-list-number + the list is a list.

Cantor creates out-of-the-list-number with n+1 real numbers, the out-of-the-list-number + the list is a list too.

With n+2 real numbers, the out-of-the-list-number + the list is a list,

With n+3 real numbers, the out-of-the-list-number + the list is a list, …

With n+m real numbers, the out-of-the-list-number + the list is a list,…

 

So, the out-of-the-list-number is a member of a list for whatever n and m. This is forever true. Cantor does not reach infinity and say “he, he, I have created a real number beyond all natural number”. No, he is always dealing with less than infinitely many real numbers.

 

Let us see the 2 case below:

Case 1: for any nÎN, real numbers are put into a list, and this is true for all members of the set of natural numbers.

Case 2: Cantor says: “for any natural number n, there is a real number that cannot be in the list”. 

 

Which of these 2 statements is true?

 

In case 2, we notice that “for any natural number n”, n is a natural number which is in the set of natural numbers; that is, n<infinity. And “there are real numbers that cannot be in the list”, the number of members of this list is smaller than infinity, which is always true in Cantor's diagonal argument.

 

So, we do not see real numbers beyond infinity and tend to believe that Case 1 is true.

 

Maybe we have to discuss that the sequence of bits of real numbers is infinite. But does infinity + 1 = infinity? Please see my paper if you wish:

Kuan Peng, 2018 «Building set and counting set»

https://www.academia.edu/37590687/Building_set_and_counting_set

 

3.       At the beginning of Sec. 2a you split the real numbers R = Sa U Sb, and then you alternately "pick" elements from each of the two sets. However, it's not clear if you mean to claim that Sa = (a1, a2, a3, ...} and Sb = {b1, b2, b3, ...}. You say "with the ai’s being the members of Sa and bi’s the members of Sb", so that does sound like every element of Sa is one of the ai's, and similarly for Sb. Again, please let me know if I misunderstand your argument. If my understanding is correct, then both Sa and Sb are countable and the equation R = Sa U Sb begs the question because the union of two countable sets is countable. (I am using the phrase "beg the question" in the formal sense of "commit circular reasoning", i.e. assume as part of a proof what one is trying to prove.) Unless I misunderstand, this would be another fallacy in the argument because you claim, without justification, that it's possible to partition R into two countable sets. If Cantor's theorem is correct, this would not be possible.

 

I do not suppose Sa and Sb be countable. We can suppose Sa = [0,0.5] and Sb =[0.5,1].

 

In fact, I put myself in the situation in the 19^th century where Cantor’s diagonal argument and the notion uncountable did not exist. This way my idea is in competition with Cantor’s but not as a destructor. When you use the terms “then both Sa and Sb are countable” and “If Cantor's theorem is correct”, you are arguing using Cantor’s diagonal argument which is supposedly true. But in 19^th century Cantor’s diagonal argument was not a correct theory and cannot be used.

 

Here is the argument that others critique me. They use Cantor’s theorem to counter me and cut the discussion without giving me the right to reply. You are the only one who discusses until this stage. So, I cannot thank you more.

 

So, we pick randomly from Sa = [0,0.5] in binary and Sb =[0.5,1] in decimal, then Cantor’s diagonal argument will not work with this alternate list.

 

4.       The same objection applies to Sec. 2b -- it doesn't matter how many countable sets you split R into, the union is still countable and by claiming that R _equals_ the union (instead of saying the union is a proper subset of R) you would be begging the question.

 

I would add to the argument above, before Cantor, there was not the notion of uncountability. All infinite sets have infinitely many members, but their cardinality is not higher than infinity.

 

Cantor introduced uncountability and cardinality higher than infinity. Uncountability is then his invention which has to be proven useful. If uncountability were useful, after 100 years, there should be many theories based on cardinality higher than infinity. But there is none, or almost none. The only use is to compare sets and decide which one is bigger than infinity. But nothing else comes, even the continuum hypothesis does not have a solution. This illustrates well that uncountability is not useful, it leads to nothing. This is a sign that uncountability may be wrong, it is in a dead end.

 

5.       Regarding your "magic box" argument in Sec. 3, you say "As the points come out forever and the demon will never run out of natural numbers, we obtain an infinite list of real numbers marked with their number of exit". There are two problems here. One is that no matter how long this process has been going on, at any point in time only finitely many points have come out. There is no time at which you have an infinite set. So what is the infinite set you "obtain" and how do you obtain it? You can only get an infinite set by making the bare assumption that it exists - no attempt at logical reasoning will allows you to derive its existence from the never-ending process of selecting one point at a time. If you don't believe that, take a look at ZF (the Zermelo-Fraenkel axioms for set theory) and you will see that one of them is the "axiom of infinity". If one were able to derive the existence of an infinite set from a never-ending process, there would be no need for that axiom. Zermelo and Fraenkel included the axiom of infinity because they recognized there was no other way to "get" an infinite set. So to proceed you need to invoke the axiom of infinity.

 

You are right, a finite process does not show the property of infinity. This is the sense of my axiome of counting infinite sets after the Magic box thought experiment, which may have the same objective as “Zermelo-Fraenkel axioms”.

 

In fact, if we have determined that a thing goes the same way repeatedly for all finite n, it will continue the same way endlessly, that is, till infinity. This is what you have done with your theorem in the email before the last email, in tiau.pdf.

Theorem. Let S be a countable set……

Proof. Since ∀n ∈ N sn ∈ [0, 1], we can express sn………….

 

You use “∀n ∈ N” which is a finite value for whatever n. But your conclusion is about infinity because you prove Cantor’s diagonal argument which is for uncountable many elements.

 

For proving Cantor’s diagonal argument you should use n = infinity, but obviously, there is not infinity.  So, Cantor’s diagonal argument could be wrong because he has not reached infinity.

 

6.       But that leads into the second problem. If the points come out one at a time, the infinite list of real numbers would be countable, so when you go on to say "the infinite list will be a one-to-one correspondence between the sets ℝ1 and ℕ", once again you are begging the question. If Cantor's theorem is correct, then the infinite list is a proper subset of R1. To disprove Cantor you need to prove that the infinite list contains all of R1. Now if I read your argument correctly, you claim to prove it by saying "each real number can get out". The fallacy here is that even if each real number can get out, that doesn't prove that each real number will get out, and as far as I can tell, there is no way to prove it. So what Cantor's theorem can be interpreted as saying about this thought experiment is that even "after" infinite time, there will still be points left in the box. In order to get them all out, you would need them to exit more than "one at a time". Even if the first point comes out at time t = 1 second, the second point at t = 1.5 seconds, the third at t = 1.75 seconds etc. (as e.g. in the "Thompson's Lamp" paradox) you have a problem at time t = 2 seconds because now the points are no longer coming out one at a time and therefore they can't be given numbers. "One at a time" means there is a time interval between each point and the next.

 

Here too, we have not the right to use Cantor’s diagonal argument. We have to see the box like a mathematician in the 19^th century, without the idea that sets can be or not uncountable, we see the points coming out regularly marked with real numbers and natural numbers. What conclusion can we draw from that?

 

7.       Going on to your second demon experiment, the conclusion "Finally, he settles that the two boxes contain the same number of points" is unjustified. The correct conclusion would be "Finally, he decides he can't tell which box is which".

 

In fact, I was too quick to go to the conclusion. I should say: he concludes that the two boxes are identical. Identical because he sees the same exit points and he sees no end and he does not see the inside. Identical boxes mean the same number of exited point. Why the same number inside? Because he sees repeatedly the same points at the exit and this will be the same process forever, which lead to my axiom. The axiom is based on the fact that infinity is not a number, but an endless process. So, what is in the boxes are not important.

 

8.       Axiom A is correct, but it only applies to countable sets, i.e. only countable sets can be "picked one by one" because the process of selecting elements of a set one by one is equivalent to counting them. The fallacy is when you say "each thing can be selected", you think that implies that "all things are selected". According to Cantor's theorem, it does not imply that, so you have denied Cantor's theorem but you have not disproven it. I think the language is misleading you -- the word "each" refers to things being taken one at a time, while the word "all" refers to the totality taken all at once. "Each" and "all" don't mean the same thing.

 

The premise of the boxes is that everything inside has the same chance to exit. Imagine that the points inside move at the speed of light, then no point can be chosen consciously like odd or even number, the point at the exit hole is there just by chance, arbitrarily. So, the points in the [0,1] really get out randomly, rational or irrational.

 

Yes, “all” and “every” mean different things. So, from the box, every point has a chance to get out, but all points do not get out.

 

I insist not to use Cantor’s diagonal argument on the reasoning because in the 19^th century the notion of cardinality higher than infinity did not exist and after this time Cantor’s diagonal argument proposed it. Imagine that I was there and I proposed that there cannot be a higher thing than infinity. In this situation, we have 2 competing ideas:

1.       Cardinality higher than infinity ; Cantor

2.       There cannot be higher thing than infinity, me

 

In this situation, you have not the right to deny idea 2 by saying that idea 1 is correct. This is because both ideas are not proven. Both ideas are on the same logical level. We have to use other ideas to prove or disprove 1 or 2.

 

Like the battle between heliocentrism and geocentrism. The church just said that heliocentrism is God’s idea and thus is absolutely correct. Then they burned Giordano Bruno. But the critical argument did not come from God, but from the planets. So, we cannot just say that Cantor’s diagonal argument is absolutely correct and reject that “There cannot be a higher thing than infinity”. So, we have to find other arguments than Cantor’s to prove or disprove the 2 ideas. So, I hope that the words countable, uncountable, Cantor should not be used in our discussion. When these words are used, the arguments are biased.

 

Instead of countable or uncountable sets, we can refer to them as infinite sets.

 

For deciding which idea is correct, we can put Cantor and me before [0,1] and let us put real numbers and natural numbers in pairs. Time can be infinitely long. As a result, he will always have a real number paired with a natural number and I will too. So, we cannot say who wins.

 

However, my idea is that when things go the same way forever, it will go the same way till infinity, which has been correct at any paired real number.

 

He states that real numbers are more numerous than natural numbers. As he does not reach infinity, he cannot show that after natural numbers are depleted, there still remains real numbers. So, he is always in the wrong. Even if he claims that he has found a real number that is not in the list already made, this real number will be paired with a natural number and he does not show his claim true.

 

In other words, we can claim that because Cantor’s diagonal argument has not reached infinity and was on the wrong side all the way, it may be not correct.

 

9.       Another way to see that your reasoning is insufficient is to note that it implies the existence of a uniform distribution on the natural numbers. We allow that the unit interval does admit a uniform distribution, with probability density function f(x) = 1. Now if that set is countable, you get a uniform distribution on the natural numbers N (look up the "de Finetti lottery" if you aren't already familiar with it). Under that distribution, all natural numbers have the same probability. So what is it? If it's zero, then the total probability is also zero (the limit of the partial sums in the infinite series). On the other hand, if it's some positive number a > 0, then by the axiom of finite additivity, eventually you get a partial sum > 1. Either way you contradict the axiom that the total probability of the sample space is 1. Some people incorrectly assume that the problem is with the axiom of couuntable additivity (which Kolmogorov included because he wanted a sigma-algebra in order to express probability in terms of measure theory) so denying countable additivity would eliminate the contradiction. But that is mistaken. The contradiction arises from finite additivity, which no one denies.

 

This probability issue is a new and interesting subject in relation to infinite sets. If I understand well, if the unit interval admits a uniform distribution, with probability density function f(x) and Integral[f(x)]dx = 1, then the probability of one point being picked is not zero because a point can be picked anyway.

 

But how does Cantor’s diagonal argument solve this problem? If for an infinite set the probability for one point is neither zero nor positive: because the probability is 1/infinity. Then for a unit interval that admits uncountable points the probability for one point is 1/infinity^2 or smaller, which is more problematic. Does this make Cantor’s diagonal argument correct? Probably not.

 

Another way to think about this issue is to imagine the set of natural numbers from which one picks a member. What is the probability for one member, given that the members are uniformly spaced in the set? This will beg the same question for the unit interval with a uniform distribution. The probability for one number to be picked is neither zero nor positive. Is this a problem?

 

In fact, I think that you have put the finger on a new discovery, that is, probability is only valid for finite sets. When we integrate the probability for a continuous function we do Integral[f(x)] dx = 1. But the value of dx is not zero, otherwise the integral equals zero. So, the function f(x) is in fact not continuous, but cut into infinitesimal segments, which have not zero length.

 

The difference between a real number point and the unit interval with a uniform distribution is that the length of the points is zero while the length of a segment dx is not. So, the unit interval with a uniform distribution is not a counterexample for my argument.

 

10.    In the beginning of Sec. 4 you say "when a sort of thing can be picked one by one, the set of infinitely many these things should be countable because picking is counting", which appears to be the basic intuition that is motivating you. It's true that whatever you have taken out of the box after infinite time must be countable, but what isn't necessarily true is that eventually every individual thing in the box will be taken out. This can happen even if the contents of the box are countable, for example the box can contain all natural numbers but you only take even numbers. After infinite time you get a countable set, but there are still things left in the box. However, Cantor's theorem is saying something even stronger: It's saying that if what's in the box is the set of real numbers (or the unit interval, which can be put into 1-1 correspondence with the entire set of real numbers via the function f(theta) = tan theta for -pi/2 < theta < pi/2) then no matter how you take the elements out, you will still not get everything.

 

Sorry to remind you that we are in a situation where Cantor’s diagonal argument does not exist. When you say “even if the contents of the box are countable” and “you get a countable set”, you still have Cantor's theorem in mind. Measuring things with Cantor's theorem troubles the reflection. Without Cantor's theorem the interval of real numbers is an infinite set, but not countable or uncountable.

 

In the box, we have the interval of real numbers, we cannot get everything out. But from the box of natural numbers we do not get everything out either. So, both boxes behave in the same manner. What is important is that when a real number gets out, a natural number gets out, we know that this is true forever with no end. The structure inside the box is not knowable. Mathematics does not deal with something that is not knowable. The thing we do with the unknown is to try to know it, not to guess what it is and use our guess to reason about it.

Letter to a reader

 Hello **,

Thank you for continuing the discussion. I see that you have written the diagonal argument into a positive proof and understand it. In fact, I have realized in the discussions with many other people that when a person uses a theorem as proof, he believes strongly that the theorem is true definitively. This belief has 2 sources:

1.       He knows how to derive it.

2.       The mainstream opinion supports it. That is, everyone says it is true.

 

When he believes strongly that the theorem is true, this person unconsciously refuses to consider that the theorem could be wrong and does not dig into the foundation of the theorem. Please do not take my thought as against you personally. I just discuss it as a psychological issue because all the people with whome I discussed showed this state of think.

 

The most common manifestation of this state of thought is to oppose my argument with his “true theorem” without considering or thinking about my argument from the ground up. When someone does not consider 1 argument and look into every step of it, he cannot see the value of the new thought and will always stay away from the heart of the proposition. In this case, the discussion is in this way:

1.       I propose a new thought.

2.       He replies: there is a theorem that says the other way. Because thousands of people have accepted it, it is definitely true. So, you are wrong.

 

In this way of discussion, he has not even seen what I have really proposed and rejected my proposition at once. So, there is no real discussion at all.

 

There is a difference between seeing a new proposition as a text/conclusion and seeing a new proposition by looking into the proof step by step and thinking about the validity of each step. For example, the step by step derivation of Cantor’s diagonal argument is:

1.       For any n in the set of natural number, we can express a real number as sn a sequence of 0’s and 1’s

2.       We can have a list of sequence of 0’s and 1’s, sn

3.       For all n in the set of natural numbers, we can create a sequence sn that is not in the list.

4.       Conclusion: for any list of real numbers including lists that contain all the real numbers, there will always be real numbers out of the lists. So, lists that contain all the real numbers cannot exist and the set R and is not countable.

 

I propose that Cantor’s diagonal argument is wrong because in this derivation, there is the step “any n in the set of natural number”. When we say “any n in the set of natural number”, n is a natural number. So, lists that contain all the real numbers cannot be represented by any list of sn, that is, for any n and any list of sn, there are real numbers that are out of the list of sn. This is not because real numbers cannot be put into a list, but because in the condition of the step 1, “any n in the set of natural number” does not allow lists that contain all the real numbers. So, the fact that there exist real numbers out of any list of sn is because the lists that contain all the real numbers is not in the above proof, because n is a finite value, but all the real numbers are infinitely many.  So, all the real numbers are not taken in the above proof which makes the proof flawed.

 

In the discussions with other people, nobody has considered that step 1 “any n in the set of natural number” means that all the real numbers are not taken in the above proof. But they all say that Cantor’s diagonal argument is true and that’s it.

 

For convincing people, I have proposed many other schemes. For example, I proposed that we can make a list whose elements are alternately expressed in binary and decimal systems. This way we cannot create the out of list real number. But they say immediately “all real numbers can be expressed in binary”. Can a binary and decimal number be exactly the same? Their limit can be the same but their sequence no. However no one discusses this aspect. In fact, not theorem proves it.

 

I have proposed another way of thinking. Any real number is a point, so, we can give each of them a name, like pi, e,… . If we can name a number pi, why not name all the real numbers by name? But they say that “one cannot name them all”. We see that their answers are instinctive without any mathematical thought.

 

Why are their answers so quick and without any dip thought? Because they have a belief in the theorem. This is a belief like a religious belief, that is, “we believe and no questioning is accepted”. This belief gives them the mental strength to give out answers like “all real numbers can be expressed in binary” and “one cannot name them all” without any proof and without any fear of being wrong.

 

This manner of thinking is a psychological issue, not a mathematical one. Belief is a psychological issue, doubt and reflection about each step of a proposition is the correct mathematical method to evaluate argument. If the argument is wrong, there is no loss. But if a new argument is correct, then this is a progress and discovery that every mathematician dreams of. But, psychological belief is an obstacle to progress, like the belief in Aristotle and geocentrism in the middle age.

 

Sorry if my reply might make you feel bad. I do not want to hurt you who kindly discuss with me. I just want to express my feelings.

 

If you are interested in the argument with the list alternately expressed in binary and decimal systems and naming all real numbers, the paper is 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

Peng Kuan

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

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Construction of the diagonal flipped number

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