|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.
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 1-9 (i.e. any number that is not zero)
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.
Well what happens when we make the next step? We need to make the list 2 digits wide now.
Here is a 2-digit list of all possible numbers represented by 2 numerals.
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.
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:
It's not square. It's twice as tall as it is wide.
Add another digit it gets worse:
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.
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 so-called "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 151: Tue Apr 23, 2019 12:22 pm
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That is irrelevant. As I said, the list cannot be 10^n times higher than wide IF IT IS INFINITELY WIDE.
Also, there is no need to make the list in ascending order, writing the smallest numbers first. Cantor's list is randomly shuffled and his first number has an infinity of digits...
Post 152: Tue Apr 23, 2019 3:09 pm
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What rule of numbers or mathematics says that I cannot add zeroes at the end of a given number? It's the same number with or without the zeroes. You say "cannot." I see no valid reason why not. The fact that adding more digits via zeroes allows additional numbers to be represented is not a valid reason. With or without the additional zeroes, and whether or not the representations is 'square,' neither the individual numbers nor the set of numbers changes.
Changing the representation does not change the underlying function between the sets.
00 -> 0
01 -> 1
10 -> 2
11 -> 3
is the same exact bijective function between the exact same two sets as
00.00 -> 0
01.00 -> 1
10.00 -> 2
11.00 -> 3
It seems to me, you continue to be confused by the ambiguous way in which you are using the term 'complete,' although I accept you can define it as you clarified below.
Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333. . . Labeling the latter as a 'calculus limit' in order to dismiss the legitimate use of the infinite decimal expansion is simply a semantic game without basis in logic.
Again, taking your clarification below into account, you still have an error in applying the finite situation to the infinite.
OK, so let's say adding the zeroes to create a finite square now means I have to make the list longer to include all possible numbers that can be represented with that many zeroes, both zero and nonzero after the decimal point, in order to make the list 'complete' in the sense you understand it.
The problem is your argument is irrelevant when you move to the case Cantor is actually considering.
The infinite list that Cantor has cannot get longer in any meaningful way. This is due to the previous point I made about infinite sets having subsets of the same cardinality.
If I add one number to Cantor's list, say the one he creates through diagonalization, the list is not really any longer. As an illustration, consider the following function.
1 -> 11
2 -> 12
3 -> 13
In one sense, the domain of this functions, all natural numbers is 'longer' having 10 elements that are not in the range (the set of natural numbers greater than 10). But the one-to-one correspondence shows they are the same cardinality, which I suppose you could say means they are the same length. Thus, adding any number of numbers to an infinite set, even adding countably many, does not change the length of the list in any meaningful way.
In fact, I already noted the function f(n)=2n gives a bijection from the natural numbers to the set of even numbers.
The function f(n)=2^n gives a bijection between the natural numbers and the powers of 2, even though you could say the powers of 2 represent in an intuitive sense an 'exponentially smaller' or 'shorter' set.
Set theory is part and parcel of this whole question. The whole point is to prove the set of real numbers has a different cardinality than the set of natural numbers. Cardinality is a concept in SET THEORY. It is defined using bijective functions between sets.
Cantor's argument is based on an assumed (for purposes of contradiction) bijective function between sets. The argument depends on the fact that real numbers have decimal expansion representations, and many (most!) numbers have infinite decimal representations. The bijective function can be considered even without considering how the real numbers might be represented.
Again, the main problem is your insistence without any logical basis in dismissing infinite decimal representations.
OK. I will accept you have now clarified that your use of the word 'complete' corresponds to my second description. Using your now clarified definition, my list is not complete. However, the question is whether your notion of complete is relevant to Cantor's proof.
Not being square does not make the list 'invalid.' I see no valid reason for this claim. Secondly, speaking of a list being square or not makes no sense with respect to infinite lists.
Here is an infinite list of integers.
I suppose you could in an intuitive sense call this list 'triangular.' But it could just as easily be made 'square' by putting in a decimal point and infinitely many zeroes afterwards for each number. 'Squareness' simply has no definite meaning with infinite lists. Even if it did, a list not being 'square' is irrelevant to Cantor's argument since each number is represented by an infinite decimal expansion.
And there is no valid reason for dismissing infinite decimal expansions, whether you use the term 'calculus limit' or not.
Your response is a nonsequitur. The sets in question have actual numbers as their elements. Cantor's argument involves a bijective function between sets of numbers. I have no idea what 'two different answers' you are accusing me of here. Two different representations of the same set still represent one set.
You have several errors here. Cantor does not 'cross digits off.' He constructs a number by selecting digit N to be different than digit N in the Nth number in the assumed (for purpose of contradiction) bijective function. He says nothing about his process resulting in a 'square.' The 'squareness' of the finite representation of his infinite list is irrelevant and nothing more than a vague intuitive notion.
Your claim about the number he creates already being on the list is faulty reasoning based on your invalid assumption that what works for finite lists of numbers with finitely many digits must necessarily apply to what Cantor is doing with an infinite list of numbers with infinitely many digits.
This again is an error on your part as what is true for finite lists of numbers with finitely many digits need not be true for infinite lists.
You really should go read the Mathematical Cranks book. It covers a lot of discussions like the one we are having. It documents how even a well-meaning amateur can go off the rails by continuing to insist that their thinking must be right while all the mathematicians must be wrong. Many of these examples exhibit the type of errors in thinking you are exhibiting here. Again, I hate to be unking, but that is the reality of this situation.
Post 153: Tue Apr 23, 2019 9:09 pm
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It has no exact decimal representation. You can obviously use decimal notion to approximate 1/3 as close as you like. However since you can never write out an infinite number of 3's or actually use that representation in any real way.
What good does it do to even claim that 0.333... equals 1/3? Other than a useless empty claim it has no practical value. And even in mathematics, when we say that 0.333... = 3 we're actually making this statement using the calculus limit as our justification. So why not just say, "In the calculus limit 0.333... = 3". To do anything less is nothing short of laziness. What you are arguing for is nothing short of lazy mathematics.
Obviously it depends on context. If you are claiming to have complete list of numbers for that numerical system then you can't just arbitrarily add zeroes without taking into consideration how that will affect the complete list.
I've clarified this several times over already. By 'complete' I simply mean every number that a numerical system can represent.
You seem to be ignoring the context of Cantor's argument. Cantor is trying to claim to be able to produce a number that isn't on the list. Well, if his list isn't a 'complete' list then his claim to have created a number that isn't on his incomplete list is utterly meaningless. He hasn't proven anything if the list he's creating is incomplete.
So that's why it's paramount in this situation and cannot be ignored.
First off, even in mathematics to say that 0.333... = 3 is nothing more than a semantic game. Give me an concrete example where this could even be used in a practical real live situation. It can't because it's meaningless to even speak of 0.333... existing in any real world sense.
So you all you are doing is arguing for a mathematics that is itself nothing more than semantic games.
Secondly, this question has absolutely nothing at all to do with the question of Cantor's diagonal proof. Even if 0.333... does equal 1/3 exactly that still doesn't help Cantor's problem.
But Cantor isn't making a bijection between sets. He's attempting to make a bijection between the set of natural numbers AND a numerical representation of decimals.
Not only this, but if he is going to claim this his "list" represents the set of decimals then we can see that his set is incomplete. Precisely because his list of decimals is necessarily incomplete because it's necessarily a square list. Therefore there should be no surprise at all that he arrives at the absurd conclusion that he is creating numbers that aren't on the list. Of course he would. That's exactly what his method much do.
So it's a false representation of a bijection of sets. His list is not the same as the actual set of decimal numbers.
Therefore if he's trying to make a bijection between sets his method fails miserably as I have been describing.
I agree. I'm not renouncing the bijective function. A bijective correspondence can indeed be made between two sets. However, in this case that's not what Cantor is doing. In this case his representation of the set of decimal expansions is incomplete. Therefore it's not a valid bijection between sets.
This is an incorrect statement of my position. I have never dismissed infinite decimal expansions. All I'm saying is the list that Cantor is construction is not a complete list of decimal expansions. Period.
So when he claims to be creating numbers that aren't on his list it's a meaningless claim. Just because he is creating numbers that aren't on his incomplete list does not mean that those numbers do not exist in the actual set of decimal expansions.
He's claiming to be creating numbers that aren't on the list. And that's his error.
Finally some progress.
Cantor's entire proof requires that "his list" contains all possible decimal expansions. Why? Because he's claiming to be producing a number that cannot be on the list. Therefore if his list does not contain all possible decimal expansions then his proof is meaningless. The fact that he can create numbers that aren't on 'his list' says nothing about the actual set of decimal expansions.
So his proof fails to show what he had hoped to be showing.
Apparently he didn't catch this, because he just assumed that lists of decimals could be listed in a square list.
Why does his list have to be square? Because he's using a diagonal line to create then numbers he's claiming are not on his list. But that method doesn't work for rectangular lists. Why? Because his diagonal cross-out-line simply can't descend down the list at a fast enough rate to cover all the possible decimal expansions. This is an innate property of the numerical representation of numbers that Cantor apparently never even gave a thought to. He was so concerned with making an abstract bijection between sets that he failed to realize that his diagonal line construction cannot be used to make this bijection.
You seem to be too hung up on the geometric idea of a square. Perhaps it's that term that is confusing you. The point is that any complete list of numbers expressed using numerical symbols must necessarily have more rows than columns. Ignoring this fact in an attempt to imagine a geometrically square list does nothing to address the core issue.
The fact that Cantor can produce numerical representations of numbers that aren't on his list is meaningless, because his diagonal line method cannot cross off more than one digit per row. And that's simply not enough to perform the task he needs.
But Cantor's list of decimal representations cannot be valid representation of the set of all real numbers.
Let's not forget that he is claiming to be creating numbers that aren't on the list (i.e. aren't in the set real numbers. Talk about something that is nonsequitur.
He does indeed 'cross digits off' why do you think they call this the diagonal line proof?
He constructs his number by crossing off digits and replacing them by his free choice of other numerals. And what other numerals is he choosing? He's choosing from numerals that make up the very notation that he is using. But he hasn't taken into account that those numerals require more rows than columns to fully describe these numbers.
He doesn't need to mention anything about "squareness". In fact, it's pretty clear that he ever even realized this folly. The squareness of his construction comes from the fact that he can only cross off one digit per row. So he's stuck with that whether he likes it or not.
And of course it follows that he'll create numbers that aren't on the list. That's naturally going to happen in any case because complete numerical lists of numbers necessarily require more rows than columns. So if you go down any complete list of numerals constructing a new number using a diagonal line that crosses off 1 digit per row you will ALWAYS produce a number that isn't on the list above where you are currently working.
In other words, this method of creating new arrangements of numerals is innate to the process he's invented. It has nothing at all to do with the actual properties of any sets. Yet he's claiming to have made a valid bijection between sets. But he hasn't done that at all. All he has managed to do is ignore the innate property of complete lists of numerals.
Pretending that this process could be legitimately taken to infinity is a grave error.
I have already shown how and why Cantor's list get's further and further behind the real list of decimals with every digit he crosses off.
If anything he's getting infinitely behind the set of reals.
How in the world can you expect to take a process out to infinity when the process gets further and further behind with ever step you take?
In order to show that something works at infinity you need to show that it works for every step on the way toward infinity and there is nothing that would prevent it from continuing.
You can't just claim that a process that can't even be made to work at all will somehow magically correct itself at infinity. Where is there any justification for that?
I showed why this thing can't even get off the ground, but you want to take it to the edge of the universe and claim that by the time it gets there everything will magically be ok.
Where's the justification for that?
If you can't even get it off the ground you aren't going to make any journey to infinity.
It's hardly a proof of anything if the only thing you can claim about it is that it will magically repair itself at infinity if we simply ignore the fact that it doesn't even work at square one.
Sorry, but even the most prominent mathematicians fully realize that there are major problems with our mathematical formalism. They discuss these issues all the time.
In fact, mathematicians can't even agree on whether our mathematical formalism is invented or discovered.
Here's a trailer of a recent 3-part documentary on precisely this topic.
If you think that mathematics is fully understood, or even truly trustworthy in all of its claims, you really need to rethink that.
I've already answered many of the questions that Hanna Fry asks in this video. And it's also true that many mathematicians don't even agree on many of the things that the mathematical community has chosen to include in the formalism.
So if you think that mathematicians are all in agreement on these things, then you need to stop reading books about cranks and start reading books about real mathematicians. You'll quickly find that they have vast disagreements on many mathematical issues.
Mathematics if far some being carved in stone. And mathematics, being human, are indeed prone to making mistakes.
Our ideas about abstract concepts such as infinity are indeed quite abstract, and there is no reason to accept that these ideas are anything more than the imagination of men.
In fact, I'm not the only one who questions Cantor's ideas. There are actually quite a few mathematicians that don't take his ideas seriously. And rightfully so. Cantor treats infinity as though it is a finite quantity. How utterly absurd is that?
Post 154: Wed Apr 24, 2019 5:11 am
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|a perfect representation of real numbers and this representation requires the zeros. It has been repeatedly pointed out to you, by the simplest arguments, that you are wrong and you refuse to accept it.Your objection is based purely on a misunderstanding of decimal notation. Those zeros exist implicitly in the notation. They are as valid as any non zero digit in a real number representation. We don't normally write the zeros because it is not practical but Cantor's method requires|
Post 155: Wed Apr 24, 2019 8:33 am
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Sorry, but your argument doesn't hold in Cantor's situation.
Why? Because Cantor is changing the zeroes to 1's in order to create his new number that he claims isn't on his list.
So his argument cannot treat those trailing zeros as being nothing more than place holders. If he's going to be changing them to 1's then he can't use your argument.
You can't change those zeroes to 1's and claim that it doesn't make any difference.
In fact, in Cantor's actual argument, since he's using base ten, he allows that those zeroes can be replaced with any digit between 1 and 9. So he's giving that decimal position the full potential.
So he cannot use your argument to defend his diagonal line argument.
You are thinking in terms of representing just one number. Cantor is talking about creating an entire and complete list of numbers. It's important that Cantor's list be a complete list. Otherwise it would be meaningless for him to claim that he is creating new numbers that aren't on his list.
After all, if his list is incomplete, then what would it matter if he could create a number that isn't on the list? That would already be a given.
If you are creating an incomplete list of numbers and you show that there are numbers that cannot be on your incomplete list why should anyone care about that?
It's paramount that Cantor's list be a complete list of real numbers. Otherwise, claiming to have created a number that isn't on the list would be meaningless.
I have shown why the list that Cantor is creating cannot be a complete list of real numbers.
Keep in mind that his entire proof is based on the claim that his list (i.e. the list he has already created by drawing his diagonal line) cannot contain the new numbers he is creating. But his list is necessarily square (i.e. contains the same number of rows and columns). This is necessarily so because that's the only kind of list he can create using a diagonal line that creates one new row per digit.
Therefore the list Cantor is creating cannot be a complete list of real decimals.
Thus his so-called "proof" fails.
So if anyone wants to claim that the natural numbers cannot be put into a 1-to-1 bijection with the real numbers they'll need to use some other method to show that this is the case because Cantor's diagonal line method doesn't cut it.
Cantor's diagonal line proof fails, and mathematicians should have recognized this by now. Shame on them.
Note: I'm not saying that the conclusion of Cantor's argument is necessarily false. He might have lucked out and the conclusion just happens to be true for some other reasons. I'm not trying to claim that the conclusion is false. All I'm saying is that Cantor's diagonal line proof fails to prove his conclusion. So mathematicians need to have this taken out of textbooks because it's a bogus proof.
If they can find some other way to show that the real numbers cannot be put into a bijection with the natural numbers more power to them. But Cantor's diagonal line proof fails to establish this to be a fact. And mathematicians need to wake up and recognize this fact.
I've just explained why it fails. So they have no excuse for not acting on this information and having Cantor's proof rejected as being a flawed proof.
They'll just have to live without this specific proof. If they can show why the natural numbers cannot be placed in a 1-to-1 correspondence with the real numbers using some other credible proof, then they can keep that conclusion. But if Cantor's diagonal line proof is all they have, then they'll even need to retract that conclusion as having been proven.