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]

So, you make a groundless and ridiculous accusation against mathematicians concerning the decimal expansion of 1/3 and yet seem unwilling or unable to provide your own decimal expansion.

I have made no ridiculous accusations against mathematicians concerning the decimal expansion of 1/3.

Are you suggesting that any mathematician thinks that decimal notation can finitely represent 1/3? Of course they don't. So they have no choice but to be in total agreement with me on this one.

Your comments on the square root of 2 would be on par with a 5th century B.C. critic of Pythagoras.

The Pythagoreans are the ones who first made this error. When they realized that irrational quantities cannot be expressed as ratios of natural numbers they should have simply looked into why that is so. Instead they didn't know what to do so they ended up just inventing a totally bogus concept of irrational numbers. A meaningless concept that no mathematicians has been able to explain ever since.

Well, I think we will leave it here. Your notions of mathematics are not only far outside of the mainstream (not a bad thing by itself) but your definitions are so vague as to be meaningless to the point that you don't seem to understand them yourself, and your claims are internally inconsistent and without a basis in logic. You are now not only denying high school mathematics, you are denying mathematics that was well known millennia ago.

You are making accusations that have no basis in reality.

I've explained exactly why no numerical system can produce a complete list of numbers in a square format. Yet you claim that my definitions are meaningless to the point where even I don't understand them. That's clearly false. I haven't done that at all. I've explained precisely what I mean.

Again, I hate to be unkind, but the notion that you have somehow discovered all on your own deep flaws that many thousands of mathematicians missed or are unaware of is not plausible given the attempted explanations you have given here. You might consider reading the following book, into which this particular thread could be included were it to publish an updated edition. I really don't know what else to say.

I see that you are clearly at a loss. You haven't even been able to demonstrate an understanding of the problems I present, and yet you are prepared to call me a crank.

That's absurd.

[Divine Insight]

In that case the same must be true for the rational numbers because we can list the rational number in precisely the same way Cantor is doing and show that we can always create a rational number that isn't on our list.

How do you know the created number is rational?

How could it be anything else?

[mgb]

[Replying to post 134 by mgb]

Actually, it is theoretically possible that the number 'not in the list' created in Cantor's diagonalization argument is rational, although highly statistically unlikely. It's theoretically possible that all of the digits going down the diagonal (or all past a certain point) are not zero, and so one could form the number not on the list by picking all zeroes past a certain point.

Very unlikely indeed. But all that has to be done here is rearrange the list so zeros won't emerge. All that needs to be proved is that such a list can exist (I'll leave that as an exercise for you!)

[mgb]

In that case the same must be true for the rational numbers because we can list the rational number in precisely the same way Cantor is doing and show that we can always create a rational number that isn't on our list.

How do you know the created number is rational?

How could it be anything else?

Here's a hint:

0.333...

0.777...

0.444...

0......

The diagonal is 0.374... and this becomes 0.485... How do you know this is rational?

[mgb]

Correction; you can have rational and irrational numbers on the list. All you have to do is write short rationals like this:-

0.3000000000000...
0.7000000000000...


Then, the diagonal will go through the zeros and turn them into ones.

[Divine Insight]

Clearly the original problem has been lost.

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 ill-defined.

But that's another topic entirely.

[mgb]

Clearly the original problem has been lost.

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 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).

[Divine Insight]

Clearly the original problem has been lost.

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 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).

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.

[mgb]

I have already demonstrated clear back in the OP why it's impossible to do what you have just claimed to have done.

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.

In decimal notation with every digit to the right you add you necessarily need 10^n more rows.

It doesn't matter. All the numbers have an infinity of digits. The zeros are only there to facilitate the drawing of the diagonal.

[Divine Insight]

It doesn't matter. All the numbers have an infinity of digits. The zeros are only there to facilitate the drawing of the diagonal.

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.[strike]0[/strike]
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.[strike]0[/strike]0
0.1[strike]0[/strike]
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.[strike]0[/strike]00
0.1[strike]0[/strike]0
0.01[strike]0[/strike]
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 cross-off 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.