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 141: Mon Apr 22, 2019 3:22 pm


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0.3000000000000... 0.7000000000000... Then, the diagonal will go through the zeros and turn them into ones. Correction; you can have rational and irrational numbers on the list. All you have to do is write short rationals like this: 

Post 142: Mon Apr 22, 2019 3:36 pm


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Let me just ask one question: Do you agree that there cannot be a numerical symbolic system that can represent every possible number in a square list? If you believe that such a system can exist please present an example of it. Otherwise, we're being distracted into areas for which I made no claims. Strawman areas. I never renounced Calculus. And yes, I do reject the mathematical community's action to invent irrational numbers when they are totally unnecessary and illdefined. But that's another topic entirely. Clearly the original problem has been lost. 

Post 143: Mon Apr 22, 2019 3:47 pm


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If I understand you no I don't. Here is a square list of rationals: 0.2000000000000000000... 0.8000000000000000000... 0.4444444444444444400... . . . Continue the list downward and across. It has as many digits across as down (an infinity of them). 

Post 144: Mon Apr 22, 2019 4:44 pm


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You haven't shown anything. All you've done is make a claim without any justification for having made it. I have already demonstrated clear back in the OP why it's impossible to do what you have just claimed to have done. You can't even do it using binary representation yet here you are pretending to have done it using decimal notation in base 10. In decimal notation with every digit to the right you add you necessarily need 10^n more rows. So your list is necessarily extremely rectangular. Even if you take it out to infinity. Then it will just be an infinitely rectangular list. Taking it to infinity can't miraculously change a rectangular list into a perfectly square list. So all you have done is show an extreme inability to comprehend the problem. 

Post 145: Tue Apr 23, 2019 3:36 am


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But the decimal expansions are infinite: 0.20000000000000000000000...with an infinity of zeros. This makes it possible to create the diagonal: 0.2000000000000000000... 0.8000000000000000000... 0.4444444444444444400... Now the diagonal is 0.204... which becomes 0.315... So there is no problem drawing the diagonal. Clearly, the number 0.315... is different from 0.2 OR 0.200000000000000...with an infinity of zeros.
It doesn't matter. All the numbers have an infinity of digits. The zeros are only there to facilitate the drawing of the diagonal. 

Post 146: Tue Apr 23, 2019 9:06 am


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You are still ignoring the fact that with every digit you add you need to add 10^n row if you want to claim that your list is complete. Since when has mathematics become a game of just making unsubstantiated claims as you are doing? Again, you don't seem to understand the problem. Try it again with the simplest example possible, binary. And we can place a leading zero and a decimal point after the leading zero if you like. We will also assume that the leading zero is just a place holder, in other words, will will never assign it a value of 1. In this way we can ignore the leading zero. In binary, if we go out just one digit we have the following complete list. 0.0 0.1 Notice that the list is already rectangular from the decimal point to the right. If we cross out the the first zero and replace it with a 1 we have a number that begins with one after the decimal place. We can claim that this number cannot be on our NEW list. However it would be wrong to claim that it isn't on the original list because it is. 0. 0.1 Our new number is 0.1 but that number is already on the list. Now let's add another digit so we can continue with our diagonal method of creating a new number. When we do this we must also add two more rows to our list. 0.00 0.10 0.01 0.11 Now we cross out the first two zeroes and claim to have a new number that is 0.11. 0. 0.1 0.01 0.11 But as you can see that number already exists. It's the fourth row down. A row that we could not have possibly gotten do by drawing a diagonal line that crosses out every new digit. Let's continue on and add yet another digit. In order to do that our list now grows by four more rows. 0.000 0.100 0.010 0.110 0.011 0.100 0.101 0.111 But we are creating new numbers by drawing a diagonal line remember? We we have the following: 0. 0.1 0.01 0.110 0.011 0.100 0.101 0.111 If we replace all these zeroes with 1's then we end up with 0.111 which we claim is not on "our" list. That's fine, the only problem is that it's obviously already on the list at row number 8. In short, because it's the innate property of numerical representations of numbers to always require far more rows than columns, the list grows exponentially faster in rows with every column we add. Therefore to claim that any new numbers we have created by using a diagonal crossoff method aren't on the list, is a bogus claim to make. It's simply false. We aren't in a position to be making such an irrational absurd claim. The situation only continues to get worse with every digit we cross off, because our diagonal line method demands that to cross off another digit we must always add a new column. So we can never truly say anything at all about what numbers might be on this list. The numbers we have created are clearly on the lists above. In fact, it would be impossible for us to ever create a new number that isn't on the list. ~~~~~~~ What you want to do is ignore this truthful fact about the nature of numerals and proclaim, without proof or reason, the if you pretend to have an infinite list it will suddenly and miraculously become square. But why should that happen? At what point will it happen? If you stop the process at any point the real list of numbers will always contain many more rows than what you were able to cross off with a diagonal line. So when does this list magically become square? Has mathematics become nothing more than a game of magic? ~~~~~~~~ 1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list. Do you disagree with this? Is so please address that point and give an example of a complete list of numerals that is square. Proclaiming that the list will somehow magically become square at infinity is nonsense. 2. I have also demonstrated that it is the nature of these numerical lists to grow exponentially in the number of rows required to record every numeral with every column appended. Do you disagree with this? If so place address that point. 3. I have shown that crossing off digits using a diagonal line that crosses off the next digit in the next row necessarily creates a square list of crossed off numerals. Therefore that list cannot be a complete list of numerals, nor can it be used to prove what may or may not be on a complete list of numerals. Do you disagree with this? Is so please explain why without resorting to the absurd claim that when taken to infinity a complete list of numerals will magically become square. Where is there any rationale for such an absurd claim? I have already shown that completed lists must necessarily grow exponentially in rows with every digit column added. Therefore there is no justification for claiming that these innately rectangular lists will magically become square at infinity. The situation only gets worse with every digit crossed off the list. What's going to magically make the situation get better as you approach infinity? You need to explain how that works. You can't just claim that it will magically become square at infinity. Mathematics doesn't stand on unsubstantiated claims, or does it? If it does, then it doesn't have much credibility. Therefore I have successfully demonstrated why Cantor's diagonal proof does not prove what he claims. 

Post 147: Tue Apr 23, 2019 10:42 am


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I think the crux of Divine Insight's error is that he/she continues to insist that the number of nonzero digits must dictate the representation be rectangular. This is an invalid assumption. There is no valid reason one cannot simply add zeroes to DI's 'rectangular' representations and still use the same set of actual numbers. Another problem invaliding DI's argument is that DI refuses to acknowledge something as basic as 0.333 . . . =1/3 or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics. He gives no valid reason for this rejection. He simply declares such notions as absurd or uses similar subjective or vague language to dismiss this. And the further problem is that he assumes things at the finite level must be equally valid at the infinite level. Again, this is an invalid assumption. I believe, for example, he refused to address that an infinite set can have propers subsets of the same cardinality while this cannot be true of finite sets. I know, I know, I said I was done with this . . .
No you have not. You have simply insisted repeatedly that your representation is the only valid one by fiat. You continue to have an ambiguous definition of 'complete' which it seems applies only to the representation, the numerals, and not the actual numbers. When I changed the representation, you insisted I had to change the set. Again, no valid basis for this claim. 00.00 01.00 10.00 11.00 is a complete list of the set (0, 1, 2, 3} expressed in binary notation. Complete, as understood in Cantor's proof, means the entirety of a set assumed to be countably infinite. If DI wants to say complete should apply to other types of sets, that is possible, but the term should mean 'the entirety of a given set' in some sense, or it has nothing at all to do with Cantor's proof, which would mean, as I showed above, DI's argument is a straw man. If DI wants to insist instead that 'complete' means all possible numbers of a given number of digits using a particular representation, that could be used, but then his argument is irrelevant to Cantor's argument. If we used base two representations instead of base 10 representations in Cantor's proof, the number of nonzero digits in some of the numbers in the list (whichever ones had a finite number of nonzero digits) would change, but the argument would be the same, just using only 0 and 1 as digits.
Your rows versus column claim only is valid if you stick to your invalid assumption that your rectangular representation scheme is the only one. If you stick to this assumption, then you are dealing with representations as the key concept, not actual sets of numbers. Any of your finite representations could be made to be square by adding a finite number of zeroes. You refuse to acknowledge this, without basis in logic or fact. It does not matter if the different representation could represent more numbers at the finite level. The different representation still represents the same set original set of numbers. The new number created by going down the diagonal will still not be in the original set. But really, the big problem is DI's refusal to accept the logical validity of infinite decimal representations.
DI labels a claim as absurd without any actual valid refutation, and the claim itself continues the 'squareness' straw man argument.. The claim that a 'square list of crossed off numbers' is created when you cross off a single number in each row is at best unclear if not nonsensical. And again, 'square' is a property of the representation, not the set of numbers itself. It is an invalid concept with respect to Cantor's proof which does depend on possibly infinite decimal representations but does not depend on this vague notion of 'squareness' which would not even seem to make sense given DI is applying it to an 'infinite by infinite' square.
DI creates the ambiguous notion of squareness at infinity and then berates mathematicians for 'not understanding' his ad hoc notion. Neither Cantor nor any other mathematicians I am aware of makes claims about 'squareness at infinity' in this context. This is DI's straw man rearing its head again.
DI has not demonstrated this, but seems unwilling or incapable of accepting that. Since DI's argument hinges on insisting without a valid reason that a representation like 00.00 01.00 10.00 11.00 must be considered invalid in favor of his rectangular representation, even though both represent the same set, his argument is an invalid straw man. Consider, for example, that while Cantor's proof typically is phrased in terms of the real numbers between 0 and 1, it could just as easily been phrased in terms of the real numbers between N=10^(1000^1000) and N+1. We just add 1 and the appropriate number of zeroes ahead of the decimal point. This would make, in the same vague sense that DI insists Cantor's list be square, the representation rectangular but wider than it is long by the number of digits in N. 

Post 148: Tue Apr 23, 2019 11:27 am


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It doesn't. I have already shown this. All one has to do is write the zeros: 0.2000000000000000000000000000000000000000000000000000000000000...so it can be made square.
They don't have to grow like this. The list can be made square by adding zeros. You are making much ado over mere notation. Adding zeros allows the diagonal to be drawn because the array is 'square'.
You are using numbers with a finite number of digits. Cantor's numbers have an infinity of digits and this fact allows him to create a new number. You cannot apply finite arguments to infinities. This is the list you gave: 0.000 0.100 0.010 0.110 0.011 0.100 0.101 0.111 and your diagonal is 111. This is what it should look like: 0.000000000000000000000000000000000000000000... 0.100000000000000000000000000000000000000000... 0.010000000000000000000000000000000000000000... 0.110000000000000000000000000000000000000000... 0.011000000000000000000000000000000000000000... 0.100000000000000000000000000000000000000000... 0.101000000000000000000000000000000000000000... 0.111000000000000000000000000000000000000000... where there are an infinity of zeros to make it 'square'. Now take the diagonal and add 1 to each digit. Your diagonal is now: 11111111... not 111
That does not matter if I am adding an infinity of zeros. 10^n x infinity = infinity.
It is not an unsubstantiated claim. It only involves the simplest rules of notation. 

Post 149: Tue Apr 23, 2019 12:13 pm


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Yes there is a valid reason. You cannot simply add zeroes in an attempt to make a rectangular list square. Every zero you add demands that your list must necessarily become exponentially more rectangular. So adding zeroes while pretending to ignore the innate property of numerical lists is not valid.
First off, both of these accusations are false. I never refused to acknolwedge that 0.333.... = 1/3 in the calculus limit. To the contrary I acknowledge this. But anyone who understands calculus knows that this does not mean that 0.333... = 1/3 in any other way. So you are misrepresenting my position on that, plus this is a total distraction from the problem at hand which you clearly do not yet understand since you still think you can just add zeros without affecting the length of the list.
Exactly. Whether an irrational relationship should be defined as a real number is a totally different topic and has absolutely nothing at all to do with the argument Cantor is making. So to even bring this up is an offtopic strawman that has nothing at all to do with the arguments put forth in this thread. Whether the square root of 2 should be defined as a real number or not has nothing to do with the problem at hand. So clearly you are grasping at straws because you obviously cannot address the points I've actually made.
Set theory has nothing at all to do with this topic. Cantor's argument is based on listing numerals and crossing off digits claiming that he has created a number that isn't on the list. But I have shown that this is a false claim on Cantor's part. Set theory doesn't even need to be mentioned at all. So again, you haven't address the points I've made. All you've done is try to distract away from them.
There is nothing ambiguous about my definition of 'complete'. Apparently you simply haven't yet understood the argument. The list you have listed above is far from complete. Look at this list: 00 01 10 11 That is a complete list of all numbers that can be represented by a 2digit binary representation. There no other possibilities. This is a complete list. Your list is far from complete. I can name 12 more valid numbers that don't appear on your list. In fact, any number you create by drawing a diagonal line down your list and replacing 0's or 1's with their opposite will necessarily be one of the 12 numbers that I can name that belong on that list if that list is said to be complete. Necessarily so. So once again, you've only demonstrated that you don't understand my first statement. Your list is incomplete, mine is not.
Hardly. Cantor's argument is an argument where he claims to be constructing a number that isn't on the list. But he can never do that because his list is necessarily square and therefore an invalid list.
Exactly. And not only would it be the same for the so called "real numbers" but it would also be the same for even the natural numbers. So this even shows more vividly that Cantor's argument is meaningless.
Well, if we get two different answers using actual numbers versus sets, then clearly set theory has its own problems. And that most certainly wouldn't surprise me in the least. If fact, mathematicians already know that set theory has problems so there's nothing new there.
It's neither unclear nor nonsensical. The list Cantor is creating is created by crossing off 1 digit per row. Therefore his resulting list can never be anything other than square. And far more importantly I have already shown that every single number he creates using this method must necessarily already be included on any complete list take out to that many columns. But I will grant you this much. Cantor's error was indeed due to his concentration on trying to create sets. And this is no doubt why he totally missed the fact that no complete list of any numerical system can possible be square. By looking at it from a pure set theoretic vantage point he couldn't see the forest for the trees. This actually shows how focusing on set theory can get us into deep trouble that we wouldn't have otherwise found ourselves in. 

Post 150: Tue Apr 23, 2019 12:18 pm


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Yes it is purely a question of notation. The list can easily be made square by adding an infinity of zeros making it as wide as it is long. Just write 1.00 as 1.00000000000000000000000000000000000000000000000... Edit: I just read the quote and see you have already made this very point. 

Last edited by mgb on Tue Apr 23, 2019 12:23 pm; edited 1 time in total 


