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.

[Divine Insight]

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.

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.

There simply is no need for any ideas of infinity greater than endlessness. Such ideas are totally bogus.

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?

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.

[mgb]

[Replying to mgb]

Can you more precisely define what you mean.

Do you mean a countably infinite set/list of integers?

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

[Divine Insight]

[Replying to mgb]

Can you more precisely define what you mean.

Do you mean a countably infinite set/list of integers?

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

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:

A completed list of binary numbers 2 digits wide:

00
01
10
11

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.

Add another digit it gets worse:

000
001
010
011
100
101
110
111

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.

[mgb]

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.

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.

[Divine Insight]

You cannot apply finite arithmetic to infinities;

In this specific proof not only can you apply finite arithmetic to this but you absolutely must apply it. If it can't even be made to work in a finite situation then it can hardly be extended to infinity.

Extending this to infinity is not going to change the underlying problem that no numerical representation of numbers can be made into a square list. In fact, we can see that in the case of finite lists the problem only gets worse exponentially with every digit we add horizontally.

Therefore the problem continues to get exponentially worse as we approach infinity.

What reasons can you possibly offer to suggest that this problem would suddenly no longer apply when we supposedly reached a list of infinitely long numerals?

Keep in mind here that I'm not arguing that conclusions about infinity might be wrong. I am simply arguing that Cantor's diagonalization proof is grossly flawed and doesn't hold water.

I actually have other reasons totally unrelated to this diagonal proof for why I reject the conclusions made by Cantor. But that's beyond the scope of this thread. In this thread I'm simply showing why Cantor's diagonal proof is grossly flawed and does not hold water.

His conclusions necessarily require that his lists are square, but they can't be square.

He obviously wasn't even aware of this flaw in his own logic. He doesn't need to demand that his lists are square. They would simply have to be square if his argument is to hold. But they can't be square. So his argument fails.

[mgb]

A square is a finite entity. You need to accept that finite arithmetic does not apply to infinities.

[Divine Insight]

A square is a finite entity. You need to accept that finite arithmetic does not apply to infinities.

This has nothing to do with geometry. Obviously you haven't understood the crux of the problem or you would have never made the above statement.

Cantor's diagonal "proof" is no proof at all until he can demonstrate that it actually works.

I have done precisely the opposite. I have shown why it cannot work in any situation at all, finite, or infinite.

You can hardly extend a principle to infinity when that principle already fails in finite situations before it even gets off the ground.

It's such an easy concept to grasp. It's difficult to understand why you can't see it. I even broke it right down to binary for you showing that there doesn't even exists a numerical system where it could work. In fact, that's how I discovered this myself. At first I saw that the argument is grossly flawed in base ten, because the numeral system for base ten is not square. So I quickly decided to reduce the problem to binary to see how it works there. That's when it became apparent that it can't even work in binary. And there isn't any numerical system smaller than binary. So it clearly cannot be made to work at all. Period.

I was right about one thing. It the situation does get better when reduced to binary, but it doesn't solve the problem. So Cantor's proof is invalid. And the fact that mathematicians haven't even discovered this extremely simple error yet doesn't speak very well of their thoroughness.

Apparently I might be the only person on planet earth who has caught this error.

What shocks me is that when I point it out to other people they STILL can't see the problem. That's pretty darn sad. Especially when I even reduced the problem to binary for them to show them the absolute simplest case.

How are you going to draw a diagonal line that cross over every possible digit in a binary number of even two places?

00
01
10
11

There are only two columns, but there are four rows!

You simply cannot draw a diagonal line steep enough to cover all four rows while advancing one digit at a time.

It's simply not possible.

And where are you going to find a numerical representation of numbers that doesn't exhibit this property. Base ten is far worse. Binary is a low as you can go. There is no such thing as a unitary numerical system. That simply cannot be made to work.

Show me how you can draw a diagonal line down the above 2-digit binary number list advancing one new digit each time and cover all four possibilities. With every new digit you cross off the list, the situation only becomes exponentially worse.

It can't be done. You shouldn't need a Phd. in mathematics to see that it cannot be done.

And if you already can't do it here in this finite situation how could you ever hope to claim that this could make sense if you tried to take it out to infinity?

[mgb]

This has nothing to do with geometry.

then why are you talking about squares?

[Divine Insight]

This has nothing to do with geometry.

then why are you talking about squares?

The numerical list is not square, it's rectangular. But those are just geometric terms we use abstractly. It really has nothing to do with geometry.

Can you not see the problem?

The list grows faster down the columns than it grows horizontally. Why bother bringing geometry into the picture when you should be able to see the problem from the examples I've given. I just use the term "square list" to bring attention to the fact that any complete list of numeral is necessarily taller than it is wide in terms of the placement of the digits in the list.

Show me how you are going to draw a diagonal line that crosses out every number on any numerical list.

There's really no need to even bring geometric labels into it. Just demonstrate how it can be done. You can't, because it can't be done. Period.

Why fight this? Do you feel some obligation to defend Georg Cantor?

[mgb]

The list grows faster down the columns than it grows horizontally. Why bother bringing geometry into the picture when you should be able to see the problem from the examples I've given. I just use the term "square list" to bring attention to the fact that any complete list of numeral is necessarily taller than it is wide in terms of the placement of the digits in the list.

Cantor's list does not have this problem because the digits are infinite progressions.