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»
Peng Kuan
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