Why Cantor's Diagonalization Proof is Flawed.

[Divine Insight]

Here is a Youtube video, under 10 minutes, of Cantor's diagonalization argument.

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 1-9 (i.e. any number that is not zero)


[strike]0[/strike]
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 2-digit 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.

[strike]0[/strike]0
0[strike]1[/strike]

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 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.

[micatala]

I will try to address your post soon. However, one thing needs clarifying.

What is your definition of "completed list?"


And I do note that you are still NOT actually addressing Cantor's proof. You are referring to square lists and completed lists. These are not part of Cantor's proof as stated.

[micatala]

Here is a completed list of square numbers. I am using a new notation here.

The cardinality of the number of black Presidents of the U.S. is represented by

111111111

The cardinality of the number of electrons in a helium atom is represented by

222222222

Etc.

Then, the completed list of numerals representing numbers less than the cardinality of the number of fingers on my two hands will be square.

111111111
222222222
333333333
444444444
555555555
666666666
777777777
888888888
999999999

There you have it. A square completed list.

Now, if you want to ignore this for now, that is fine, but it seems to me my argument here is quite as logical as yours. Your argument that 'completed lists' of numbers need to be far taller than they are wide does not really hold up.

ANd more importantly it is irrelevant to Cantor's proof. For one, you are not even taking into account finite decimal numbers, never mind infinite ones.

[Divine Insight]

Here is a completed list of square numbers. I am using a new notation here.

The cardinality of the number of black Presidents of the U.S. is represented by

111111111

The cardinality of the number of electrons in a helium atom is represented by

222222222

Etc.

Then, the completed list of numerals representing numbers less than the cardinality of the number of fingers on my two hands will be square.

111111111
222222222
333333333
444444444
555555555
666666666
777777777
888888888
999999999

There you have it. A square completed list.

Now, if you want to ignore this for now, that is fine, but it seems to me my argument here is quite as logical as yours. Your argument that 'completed lists' of numbers need to be far taller than they are wide does not really hold up.

ANd more importantly it is irrelevant to Cantor's proof. For one, you are not even taking into account finite decimal numbers, never mind infinite ones.

The list you have proposed is NOT a "completed list" in terms of listing all possible integers or all possible real numbers.

I can't even believed that you proposed such a thing. The list you have proposed has nothing to do with the problem at hand at all.

Moreover, use Cantor's diagonalization method on the list you have just created and you'll come up with a number that is NOT on your list! So now what do you do? Just proclaim that you don't care about that number because it's not supposed to be on your list?

Cantor's entire proof relies entirely on the idea that he can supposedly come up with a number that's NOT on his list thus showing that there is a PROBLEM.

But in your so-called "Completed List" there would be no problem at all if you came up with numbers not on your list because who cares?

Your example here totally misses the problem entirely.

I can't believe that you even proposed it.

[olavisjo]

.

Here is a completed list of square numbers. I am using a new notation here.

I think DI is very correct here.

If you list all the integers of length n, your list will have 10ⁿ members.

If n = 1 then {0,1,2,3...9} will have 10 members.

If n = 2 then {00,01,02,03...99} will have 100 members. etc, etc...

So the ratio of length to members is n/10ⁿ then as n->∞ the ratio approaches zero.

So you would have a rectangle of zero length and you are asked to find a diagonal to create a new number that is not on the list. Can you see how this would be impossible.

The proof is fallacious.


Cantor is creating a number that is not on the list that contains all numbers. This is a contradiction.

Consider this, a complete list of all natural numbers. {1,2,3...∞}

I contend that this list is not complete, since I can find a number that is not on that list. {1,2,3...∞,∞+1}

Therefore, as Cantor did, I conclude that the natural numbers are not countable.

This is Cantor's flaw, he takes the numbers out to infinity and then he adds a new member at the infinith place, and we all know you can't add to infinity.

∞+1=∞
∞-∞+1=∞-∞
1=0

[lao tzu]

I will try to address your post soon. However, one thing needs clarifying.

What is your definition of "completed list?"

Dear Micatala,

There is also the question of what he means by "square," as his usage cannot be reconciled with the object known to geometry. His "square" has infinite width and depth when he uses it to describe Cantor's list. His "square" includes its own diagonal when he claims the diagonal length of √2 of a unit square is self-referential, despite the fact the square and the diagonal have only two of their infinite number of points in common.

Despite assertions to the contrary, nothing is defined here except in contradiction to established definitions. To date we have assertions that the undefined object known to DI as a "completed list" has the contrarily-defined property of being "square."

And I do note that you are still NOT actually addressing Cantor's proof. You are referring to square lists and completed lists. These are not part of Cantor's proof as stated.

This is, of course, the only relevant point. Unless DI can address Cantor's proof, he has no hope of showing it is flawed.

As ever, Jesse

[Divine Insight]

Consider this, a complete list of all natural numbers. {1,2,3...∞}

I contend that this list is not complete, since I can find a number that is not on that list. {1,2,3...∞,∞+1}

Therefore, as Cantor did, I conclude that the natural numbers are not countable.

This is Cantor's flaw, he takes the numbers out to infinity and then he adds a new member at the infinith place, and we all know you can't add to infinity.

That's exactly right. Cantor is trying the infinite set of natural numbers as though it is somehow finite to begin with. Cantor is imagining in his mind "Completed Infinities". This is the entire basis behind all of Cantors arguments. If a mathematician does not accept the concept of "Completed Infinities" then he or she should not be taking Cantor's work seriously.

This was the objection of many mathematicians back in Cantor's day. And they never gave in to his absurdities. Leopold Kronecker never accepted it, nor did Henri Poincare.

In fact Poincare said, "Cantor's transfinite set theory is a disease from which the mathematical community will someday be cured."

I totally agree with Poincare. The only thing that leaves me in total shock and amazement is how long it's taking the rest of the Mathematical Community to recognize what Poincare saw immediately.

As a modern mathematician today I totally agree with Henri Poincare. I see the same absurdities.

If "Contradiction by Absurdity" is worth anything at all in Mathematics, then clearly Cantor's entire empty set theory is a logical contradiction because it absurdly demands that there exist 'larger and smaller' sizes of infinity.

And that is indeed utterly absurd. Infinity simply means "endless" and nothing can be more endless than endless.

It's an utterly absurd formalism that Cantor has created. And therefore it should itself be seen as a contradiction by absurdity.

[lao tzu]

Cantor is creating a number that is not on the list that contains all numbers. This is a contradiction.

Dear olavisjo,

The above is flawed in that it includes the incorrect assumption that all numbers are listable. That is the very point of Cantor's proof: It is not possible to create a list of all real numbers. His proof is elegant in that he finds a minimal contradiction, a single number, created by construction relative to any proposed list, that cannot be on that list. In fact, there are an uncountable number of such non-listable real numbers.

Consider this, a complete list of all natural numbers. {1,2,3...∞}

I contend that this list is not complete, since I can find a number that is not on that list. {1,2,3...∞,∞+1}

Therefore, as Cantor did, I conclude that the natural numbers are not countable.

This is Cantor's flaw, he takes the numbers out to infinity and then he adds a new member at the infinith place, and we all know you can't add to infinity.

Your "natural numbers" include a "number" that is not a natural number. Having included this non-natural number, you then proceed to show that still more non-natural numbers are not on your list of "natural numbers," and hence your "natural numbers" are not countable.

But this "proof" is as trivial as it is spurious. If you are free to redefine the natural numbers to include whatever additional numbers you wish, you may as well include all of the real numbers, redefining them as "natural numbers" as well, and thus showing the "natural numbers" are, like the real numbers, uncountable.

More pointedly, the "numbers" you are presenting are simply the ordinal numbers, in which ∞ and ∞+1 occur naturally, though we use lower case omega in place of ∞ in practice. But, even within the infinite cardinal numbers, you are incorrect in stating that we cannot perform addition. We can do so in a well-defined manner, and also multiplication, exponentiation, and even create factorials. It is subtraction which is generally disallowed, along with division, and the use of either additive or multiplicative inverses.

As ever, Jesse

[keithprosser3]

I'm sorry I can't join in, but writing out this **** infinite square of numbers is taking longer than I thought and I keep having to run out and get more paper.

[Divine Insight]

I will try to address your post soon. However, one thing needs clarifying.

What is your definition of "completed list?"

Dear Micatala,

There is also the question of what he means by "square," as his usage cannot be reconciled with the object known to geometry. His "square" has infinite width and depth when he uses it to describe Cantor's list. His "square" includes its own diagonal when he claims the diagonal length of √2 of a unit square is self-referential, despite the fact the square and the diagonal have only two of their infinite number of points in common.

Despite assertions to the contrary, nothing is defined here except in contradiction to established definitions. To date we have assertions that the undefined object known to DI as a "completed list" has the contrarily-defined property of being "square."

And I do note that you are still NOT actually addressing Cantor's proof. You are referring to square lists and completed lists. These are not part of Cantor's proof as stated.

This is, of course, the only relevant point. Unless DI can address Cantor's proof, he has no hope of showing it is flawed.

As ever, Jesse

I've already shown that it's flawed in more than one way.

I do agree though that my original use of the term "Square" was not well thought out. I actually came up with that term because math books often show Cantor's proof using square patterns of numbers. Like the following example from Wikipedia:



I recognized that there is a problem with this. So I started using the term "square" lists. But in truth, it only needs to be a "triangular" list. We can remove all of the numerals above the diagonal line because they are truly unnecessary.

But what is absolutely necessary is that the list take on the form of a triangle. And by that I simply mean that ever new row must contain a at least one new digit. This is absolutely mandatory in order for the diagonal line method to even work.

So the list must be "Triangular". In fact, it must necessarily be a "Right Triangle", assuming that all the rows are lined up at the left-hand side. Why is this? Well, it's because every new row must contain a new digit. The diagonal line must be at an angel of 45 degrees. In other words, it must cross over one new row and one new digit consistently. Assuming the rows and digits are square font (which is really mandatory because changing this wouldn't help the problem anyway it would only serve to confuse the issue dramatically)


So whether we talk about it as being a "Right Triangle" or a "Square" is irrelevant. All we're doing when we recognize that we at least need a Right Triangle is recognizing that we could arrange to have the numerals above the diagonal line be irrelevant.

But completed lists of numbers cannot be a Right Triangle. They necessarily must be far taller than they are wide.

It is true that they aren't exactly "rectangular". They are only rectangular until we add another digit to the width. They they are a "wider" rectangle for 10^n rows.

Then we add another digit and they get one digit wider for another 10^n rows.

So overall any "completed lists" are ultimately "Triangular" where their height is extremely greater than the width of their base. In other words, they cannot even be Right Triangles. A Right Triangle (constructed by rows and digits of the same height and width in this manner must have a height equal to it's base. And this is what I mean by it being "Square". In other words, just any right triangle won't do, we need a right triangle that is basically a "Square Right Triangle". In other words it's hypotenuse is 45 degrees.

And that's what I mean when I say that Cantor's list is "Square". The slope of h is diagonal line is fixed at 45 degrees. In other words, he can't descend the list any faster than one digit and one row at a time. Thus forcing his construction to always be "Square" in this sense.

In other words, the height of his constructed list will ALWAYS be precisely the same as the width of his list. This will never change an any point.

Pretending to carry this out to infinity is nonsense.

Before he could do that he would need to give an example of how it could work in a finite situation. Then he could extend the idea from there. But this method can't work in a finite situation. And it can't magically start working in an infinite situation either.

Henri Poincare was right. Cantor's completed infinities and transfinite numbers are a disease that the mathematical community will someday recover from. Unfortunately it doesn't appear that it's going to be anytime soon.

The "logic" of Cantor's arguments is extremely flawed. His very conclusion is an absurdity.

[Divine Insight]

I'm sorry I can't join in, but writing out this **** infinite square of numbers is taking longer than I thought and I keep having to run out and get more paper.

But there is no need to take it out to infinity.

All you need to do is list the first single-digit numerical representations of the first 9 natural numbers.

1
2
3
4
5
6
7
8
9

Then ask yourself, "Can I run a diagonal line down this list to create a new number that is NOT already on this list?"

If the answer is ALREADY no, then why trouble yourself with trying to to take this out to infinity?

You can already see that the process is bogus and can't even work in the simplest finite situation.

If it can't be done in a finite situation then how on God's Green Earth could it be extended to infinity?