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? 



