First of all, apologies if I make any grammar or vocabulary mistakes as English is not my native language.
In this post, I will demonstrate that Cantor's diagonal argument, which is the best known and accepted as the most elegant demonstration that Reals are not countable, i.e. that the cardinality of the Reals set is greater than that of the Naturals, is actually wrong. I'm not 100% sure about his conclusion being wrong, but the process he follows to reach it certainly is. It may look elegant, but it is wrong and misleading.
For those of you that are not familiar with this demonstration or have long forgotten the details, here is a link to the wikipedia page about it. From now on, all my references to the Cantor's demonstration will refer to the way it is explained in that wikipedia page.
All of Cantor's demonstration is based on, no matter which function f we have created between the naturals and T (defined as the set of all possible infinite sequences of 0s and 1s), if he can find one sequence of 0s and 1s that would not exist in the image of the function, then it follows that the function is not bijective. And he then goes on to provide a method for finding such a sequence. And he says: given that I have been able to find such a sequence, the function cannot be bijective.
Now, this is the key: the method he provides to find such a sequence is fatally flawed.
In order to find his s0 sequence of zeroes and ones, Cantor's method consists on establishing a small difference with the images of each and every natural number. With each and every sn. In particular, for a generic sn, he establishes the difference in the nth digit of the sequence. Now, the obvious problem is that in order to actually find that sequence, he needs to do an infinite number of comparisons with an infinite number of terms of which he knows nothing a priori, as the sequence of sequences of 0s and 1s does not need to be in any specific order. And he has not demonstrated that it is possible to actually do such a thing in order to actually find such a s0
sequence, when the number of operations required to do such a thing is literally endless.
Well, it may be an interesting concept, but it is unreal, and I can demonstrate that such a thing cannot be done, by providing another absolutely clear example. By following Cantor's way of reasoning, I would be able to demonstrate that the Naturals themselves, are not countable! Which is obviously wrong. But I could show it to be right only by accepting that I can find a number for which I need to do an infinite number of operations. Given a sequence of ALL natural numbers, I will show the way to obtain a natural number which does not belong to the sequence. Absurd? Well, if it's not absurd for Cantor...
Cantor goes one by one through all the sequences of 1s and 0s establishing a difference with each of them, and in each individual step of that process, he has certainly found a sequence that belongs to T and is different from all sequences already examined. That's not enough to claim that, after finishing with the infinite ammount of them, he will still have a sequence that belongs to T and is different from all others, even if such a thing is true for any finite number of sequences. In the example I'm about to provide, for a sequence of all natural numbers, I will show exactly the same. In each step of the process, I will have found a number which is certainly natural and is certainly different from all the naturals already examined. If the fact that this is true for any finite number of naturals examined allowed me to say that it is also true for the infinity of them, then by Cantor's reasoning I would have found that the naturals are not countable. And that's why Cantor's argument is wrong.
Let Sn be my sequence of ALL naturals, all different. How will I calculate a natural (S0) that doesn't belong to the sequence? Like Cantor, I am going to do so in steps, each of them involving one natural of the sequence and one operation. I will call S0,n the nth step to calculate my final S0. Cantor does exactly the same, only for him, his s0,n is also the nth digit of his s0 final sequence, but anyway it is a term that requires a calculation, and he cannot get his final s0 without that calculation.
For me, S0,n = max (S0,n-1 -1; Sn) +1. Or in other words, in each step, I check if the next Natural involved is equal or bigger than my previous result, and in case it is, my new result will be equal to the new Natural involved, plus 1. Otherwise I keep the same result as in the previous step.
With this procedure, for ANY finite n number of Naturals examined, I will certainly have found a Natural that is different (bigger) from all of the Si of the sequence, with i <= n. Now if, from this, like Cantor does, I could conclude that the same is true for the infinity of terms that my infinite sequence contains, I would have described a way to find one Natural number which is bigger that ALL of the naturals of this infinite sequence which was meant to contain them all. I would have found one natural which is bigger than all others in the sequence. So this natural I would have found would not exist within my sequence. And therefore by Cantor's reasoning, no function N -> N could be bijective.
WRONG! You cannot claim such a thing! You cannot pretend that you can find a number which, in fact, you cannot find, because finding it requires an infinite number of operations and is, therefore, impossible! Infinite means it never ends! I'm sorry Cantor, but life on earth and the Universe itself would have long ended before your method could find s0. That's because to establish a difference with every sequence of the infinite number of them, you also need to do so with the LAST one. And there is not such a thing as a last element in an infinite sequence. You can do NO OPERATION THAT REQUIRES THEM ALL. You can only do that with a finite number of terms. You cannot with an infinite number of them, unless the sequence trends towards some value known in advance and in a particular way, and this is not the case that you are dealing with. It's, actually, like trying to find a natural number that is bigger than all other natural numbers. Impossible. You cannot claim that such a thing exists, like you do. It doesn't.
And here ends my demonstration that Cantor's diagonal argument is wrong, and it is so by accepting a wrong premise, that his announced s0 does, in fact, exist.
I will appreciate any comments that you can give about this demonstration. Thanks a lot.