Posted: Thu Oct 24, 2013 6:32 pm Why Cantor's Diagonalization Proof is Flawed.  
Ok, I've seen this proof countless times. And like I say it's logically flawed because it requires the a completed list of numerals must be square, which they can' t be. ~~~~ First off you need to understand the numerals are NOT numbers. They are symbols that represent numbers. Numbers are actually ideas of quantity that represent how many individual things are in a collection. So we aren't working with numbers here at all. We are working with numeral representations of numbers. So look at the properties of our numeral representations of number: Well, to begin with we have the numeral system based on ten. This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. How many different numbers can we list using a column that is a single digit wide? Well, we can only list ten different numbers. 0 1 2 3 4 5 6 7 8 9 Notice that this is a completed list of all possible numbers. Notice also that this list is not square. This list is extremely rectangular. It is far taller than it is wide. Let, apply Cantor's diagonal method to our complete list of numbers that are represented by only one numeral wide. Let cross off the first number on our list which is zero and replace it with any arbitrary number from 19 (i.e. any number that is not zero) 1 2 3 4 5 6 7 8 9 Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it. Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using a diagonal line because the list is far taller than it is wide. Now you might say, "But who cares? We're going to take this out to infinity!" But that doesn't help at all. Why not? Well what happens when we make the next step? We need to make the list 2 digits wide now. What happens? Here is a 2digit list of all possible numbers represented by 2 numerals. 00 01 02 03 04 05 06 07 08 09 11 12 13 14 15 . . . 95 96 97 98 99 What happened? Well, our completed list of possible numerals that is two digits wide has incrusted in vertical height exponentially. This list is now 100 rows tall and only 2 column wide. Now let's cross off the first two digits of our list and replace them with arbitrary numerals. 0 Ok, for the first zero being stuck off the list, I'll chose to arbitrarily replace that with a 5. For the second digit being struck off the list I'll replace that arbitrarily with a 7. My new number is 57. Is 57 already on my completed list? Yes. It's just further down the list where I couldn't possibly reach it by drawing a diagonal line. Now you might say, "But who cares? We're going to take this out to infinity!" But duh? We can already see that in a finite situation we are far behind where we need to be, and with every digit we cross off we get exponentially further behind the list. Taking this process out to infinity would be a total disaster. You could never claim to have "completed" this process because you can't move down the list fast enough using a diagonal line that crosses off each digit diagonally. The very nature of our system of numerical representation forbids this. You can't complete this process in a finite situation, and it gets exponentially worse with every digit you add to the width, then you could never claim to have completed this process by claiming to have taken it out to infinity. "Completed Lists" of numerical representations of numbers are NOT SQUARE. Yet Cantor claims to be creating a "Completed List" here. It's a bogus proof that fails. Cantor didn't stop to realize that our numerical representations of number do not loan themselves to nice neat square competed lists. And that was the flaw in his logic. By the way you can't even do this using binary representations of numbers. In Binary Representation A completed list of binary numbers 2 digits wide: 00 01 10 11 It's not square. It's twice as tall as it is wide. Add another digit it gets worse: 000 001 010 011 100 101 110 111 There is no way that a completed list of numbers can be represented numerically in square lists. Yet Cantor's diagonal argument demands that the list must be square. And he demands that he has created a COMPLETED list. That's impossible. Cantor's denationalization proof is bogus. It should be removed from all math text books and tossed out as being totally logically flawed. It's a false proof. Cantor was totally ignorant of how numerical representations of numbers work. He cannot assume that a completed numerical list can be square. Yet his diagonalization proof totally depends on this to be the case. Otherwise, how can he claim to have a completed list? If he's standing there holding a SQUARE list of numerals how can he claim that he has a completed list? Yet at what point does his list ever deviate from being square? It never deviates from being square. It can't because he's using a diagonal line to create it. That forces his list to always be square. Georg Cantor was an idiot. He didn't even understand how numerical representations of numbers work. His socalled "proof" doesn't prove anything. It's totally bogus. He can't claim to have a "completed list" by the way he is generating his list. Claiming to take this out to infinity doesn't help. With every new digit he creates he falls exponentially behind where he would need to be to create a "Completed List". Yet that's what he claims to have: A Completed List. It's a bogus proof, and I'm shocked that no mathematicians have yet recognize this extremely obvious truth. They keep publishing this proof and teaching it like as is it has merit when in fact it's totally bogus. 
Post 81: Mon Oct 29, 2018 8:46 am


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Really, DI, you seem to be objecting to Cantor's diagonalization argument not working on finite lists of numbers with finitely many digits. You are right in this sense. The argument does not work on a finite list of INTEGERs. It would not even work on an infinite list of integers, mainly for the reason that integers ARE countable not uncountable.  
Post 82: Mon Oct 29, 2018 12:17 pm


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Well don't let the decimal point fool you into thinking anything would change just because you are working to the right of a decimal point. The rectangular property of this numerical style of notation still holds. So if this is Cantor's argument for why the reals cannot be placed in a onetoone correspondence with the natural numbers, then his arguments fails miserably. All he has managed to do here is create a false quantitative idea of larger and smaller cardinal infinities. There simply is no need for any ideas of infinity greater than endlessness. Such ideas are totally bogus. 

Post 83: Tue Oct 30, 2018 3:22 pm


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No, it does not. With an integer, say 27, adding zeroes after the 7 changes the value of the number represented. I suppose if you wanted to, you could add zeroes before the 2 and write all integer numerals as infinitely many zeroes followed by the usual representation. With decimal numbers, you can add infinitely many zeroes (or any finite number of zeroes) after the last nonzero digit and not change the value of the number. That is why your 'square' criterion is irrelevant.
Just remove the 'false' from this statement and you have it right. Cantor's proof does establish that the cardinality of the set of natural numbers is different than the cardinality of the set of real numbers. Cardinality for infinite and finite sets can be defined using the concept of bijective functions (onetoone correspondences). The set of numbers represented on the side of a standard die is the same as the cardinality of the set of sides of a regular hexagon because they can be put into one to one correspondence. The function f(n)=2n establishes that the natural numbers and the even natural numbers are of the same cardinality. The fact that the natural numbers cannot be put into onetoone correspondences with the real numbers between 0 and 1 is established via Cantor's proof. To prove the 'cannot,' we assume, for purposes of contradiction that we can, which gets us to the list which you keep insisting has to be square but which is really just an infinite list of decimal representations of infinitely many digits each. 'Squareness,' however defined is not mentioned in the proof, nor is it relevant.
Define "need." I think I asked this (or something like this) before. Do you dispute the the decimal number 0.333 . . . infinitely repeating threes is a valid representation for the rational number 1/3? 

Post 84: Fri Apr 19, 2019 2:40 pm


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But it would work on an infinite list of infinite integers; integers with an infinity of digits... 

Post 85: Fri Apr 19, 2019 2:41 pm


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But it would work on an infinite list of infinite integers; integers with an infinity of digits... 

Post 86: Fri Apr 19, 2019 4:59 pm


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Replying to mgb]
Can you more precisely define what you mean. Do you mean a countably infinite set/list of integers? [ 

Post 87: Fri Apr 19, 2019 5:43 pm


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You are wrong. It is relevant. We aren't just talking about adding zeros, we're talking about adding any digits between 0 and 9. Moreover if you truncate them at any point then Cantor's argument fails. It only appears to potentially work if you allow the lists to continue on infinitely. But the moment you pretend to do that you lose sight of the reason why it doesn't work. And the reason is precisely because Cantor's list isn't square. The bottom line is that Cantor's argument relies upon the claim that the numbers he requires are NOT on his list. But that claim makes absolutely no sense at all if the list isn't square. So the squareness of the list is paramount to Cantor's argument whether he realized this or not. And apparently he didn't realize the importance of this. And apparently no one else caught it but me. That doesn't say much for the entire mathematical community. They had over a century to discover this error and haven't even found it yet. Even when I point it out to them.
Yes, I reject this notion. This notion is just as absurd as Cantor's ideas. This notion wrongfully assumes that you could complete an infinite process. But you can't. And therein lies the fallacy of claiming that 0.333... equals 1/3 exactly. Does it qualify as being the same as 1/3 in the context of the Calculus Limit? Yes, but that's a totally different concept from saying that it's actually the same quantity. Cantor was wrong. And until mathematicians realize this they will be doomed to follow his errors for their own eternity. They are following a grave error in mathematical reasoning. And they will continue to go down this abyss of errors until they wake up and realize the grave mistake they have fallen for. Cantor's ideas about infinity are nonsense. He was wrong. Pure and simple. Mathematicians have become like stubborn theists. They treat mathematical formalism as though it's some kind of religious dogma that cannot be questioned. But in truth Cantor's ideas about infinity are not only wrong, but they are causing mathematicians to go astray from the true nature of the quantitative properties of the actual world in which we live. 

Post 88: Sat Apr 20, 2019 5:53 am


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Yes. The same procedure would create an integer not on the list if the integers had an infinity of digits. 

Post 89: Sat Apr 20, 2019 9:02 am


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Exactly. In fact, the list doesn't even need to be infinite. I demonstrated this in the OP using the simplest possible example in binary:
If we try to run a diagonal line down this simplest finite list of numerals from zero to three we instantly find that we would need to claim that the numbers two (10) and three (11) are not on the list, but they are on the list. What happened? Well a complete list is not square it's necessarily rectangular. Therefore we can never cover all the numbers on a list by drawing a diagonal line. So Cantor's method of trying to draw a diagonal line to cross off numbers will ALWAYS produce numbers that he will claim are not on the list, but actually are on the list. It works on this very simple finite list above. We can clearly see why this method is meaningless and why the logic behind it is flawed. There is no point in trying to take this process to infinity when we can see that it is already failing in a finite case. It's only going to get worse the further out we take it.
By adding one more digit we end up with 5 numbers that we claim are not on the list, when in fact they are. It only gets worse with ever digit we add. So as we head out to infinity the problem is getting exponentially worse with every additional digit. Note: I'm using binary here to show the absolute simplest case. It cannot be implied any further. There is no such thing as a square list that can represent all numbers. When we move up to base 10 things get far worse. So Cantor's idea fails entirely. There is no way to salvage it. There is no meaningful numerical representation of numbers below binary. So it's not even possible to make numerical lists of numbers that are square. It simply cannot be done. 

Post 90: Sat Apr 20, 2019 11:17 am


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You are missing Cantor's point. He is dealing with the infinite expansion of real numbers. You cannot apply finite arithmetic to infinities; eg. 1 + 10 = 11 but 1 + Aleph Null = Aleph Null. 



