Why Cantor's Diagonalization Proof is Flawed.

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Why Cantor's Diagonalization Proof is Flawed.

Post #1

Post by Divine Insight »

micatala wrote: 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? :-k

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.
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Post #131

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mgb wrote: To be honest I have no idea what your are talking about. What's this square business about? What do you mean by a 'complete' list? Why are you crossing out numbers?
You need to explain your concept again.
Ok, I can see that this is all too far advanced.

You know that numerals are not numbers right?

Numerals are simply the symbols we use to represent numbers. And this includes the decimal numerals 0-9.

I reduced this to binary to try to simply the problem, but apparently even that simplification isn't simple enough.

Binary is the simplest possible numerical system that can be used to represent numbers. But even in it's most elementary form it's rectangular. For example the simplest possible case is to list all possible binary numbers as a completed list.

Thus we have

0
1

That's it. That's the complete list of all possible numbers that can be expressed using binary notation. But even that list is already rectangular. It's column is only 1 digit wide, but it requires 2 rows to list all possible numbers.

And it's gets worse with every additional digit.

00
01
10
11

Now we have 2 column and 4 rows. In fact the number of rows will always be 2^n where n is the number of columns. So it's only going to grown exponentially.

It's a complete list in the sense that it lists every possible number this numerical system can represent. There are no numbers

I crossed out the numbers diagonally to show why Cantor's method doesn't work.

If I go down this simple binary list that is only 2 digits long crossing out digits and replacing them with alternative digits I can always create a number that isn't one the lest that I have crossed out thus far. However, I cannot say that this number does not exist on the list somewhere further down where my diagonal method can't reach.

And yet that is precisely what Cantor's proof requires.

In short, the fact that he is creating numerals that aren't on his crossed out list is meaningless because the fact that they aren't on his list is irrelevant. His list cannot be a complete list. His list is necessarily square. But there is no such thing as numerical system that can express all possible numbers in a square list.

So Cantor's list cannot be a complete list of all possible numbers.

Moreover, you can also see this another way entirely.

Just stop Cantor at any point in time and look at the number he claims to have constructed that is not on his list. Ask yourself, if that number exists. Of course it exists. In fact, not only does it exist but it's also necessarily a rational number.

So at no point does Cantor ever create anything more than a rational number. So while he claims to have created some magical new number he hasn't.
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Post #132

Post by Divine Insight »

mgb wrote:
Divine Insight wrote:The square root of 2 is an irrational relationship between two quantities that should have never been recognized as a "number" in the first place because it obviously isn't.
Well, if you are going to argue that numbers are not numbers you can "prove" anything I guess.
Mathematicians have already proven that the square root of 2 is not a number. That's when they invented the absurd concept of irrational numbers.

In short, they just arbitrarily chose to call this relationship between quantities a "number".

So it's actually the mathematicians who just arbitrarily call things "numbers" in spite of the fact that it has violated every definition of number that had existed prior to that time.

One thing for certain, they have no choice but to confess that the square root of 2 is not a "Natural Number". And why is that? Because they invented the concept of an irrational number, which was yet another grave mistake.

So at least they can't argue with me about that. Even they have no choice but to confess that the square root of 2 is not a natural number.

The only difference is that I know why it's not a natural number, and the mathematician don't. They have no clue. And they even know that they have no clue.

In fact, here's a question for you,...

What causes an irrational number to be irrational?

In other words, why is the square root of 2 irrational?

And I don't just mean because it cannot be represented by a ratio of natural numbers, we already know that. I'm asking you to tell me why it cannot be represented by the ratio of natural numbers?

Can you answer me that?
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Post #133

Post by mgb »

DivineInsight wrote:
Divine Insight wrote:If I go down this simple binary list that is only 2 digits long crossing out digits and replacing them with alternative digits I can always create a number that isn't one the lest that I have crossed out thus far. However, I cannot say that this number does not exist on the list somewhere further down where my diagonal method can't reach.
That's the problem, you are creating finite lists and of course it doesn't work for finite lists. Cantor's list is infinitely wide and infinitely high. This is infinitely different from what you are doing.

Ask yourself this: Is his diagonal number the first number? No because its first digit is different. Is it the second? No, because the second digit is different. Likewise for all infinity, it differs from all the numbers on the list.
His list cannot be a complete list. His list is necessarily square. But there is no such thing as numerical system that can express all possible numbers in a square list.
What has 'square' to do with it.
So Cantor's list cannot be a complete list of all possible numbers.
That is not the point. The point is that if his list has cardinality Aleph Null it can't be a complete list. It doesn't have to be complete, it has to be infinite. There's a difference.
Moreover, you can also see this another way entirely.

Just stop Cantor at any point in time and look at the number he claims to have constructed that is not on his list. Ask yourself, if that number exists. Of course it exists. In fact, not only does it exist but it's also necessarily a rational number.
The point is it does not exist at the point where he is stopped or before that point. But Cantor does not stop so it does not exist anywhere.
So at no point does Cantor ever create anything more than a rational number. So while he claims to have created some magical new number he hasn't.
At no point. But we are not talking about finite points we are talking about inifinity. That's the difference.

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Post #134

Post by mgb »

Divine Insight wrote:In fact, here's a question for you,...

What causes an irrational number to be irrational?

In other words, why is the square root of 2 irrational?

And I don't just mean because it cannot be represented by a ratio of natural numbers, we already know that. I'm asking you to tell me why it cannot be represented by the ratio of natural numbers?

Can you answer me that?

No I can't. I await your answer. I presume you are going to say because it doesn't exist. But what does it mean for a number to exist? As far as I can see it is enough for them to exist abstractly.

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Post #135

Post by Divine Insight »

mgb wrote:
DivineInsight wrote:
Divine Insight wrote:If I go down this simple binary list that is only 2 digits long crossing out digits and replacing them with alternative digits I can always create a number that isn't one the lest that I have crossed out thus far. However, I cannot say that this number does not exist on the list somewhere further down where my diagonal method can't reach.
That's the problem, you are creating finite lists and of course it doesn't work for finite lists. Cantor's list is infinitely wide and infinitely high. This is infinitely different from what you are doing.

Ask yourself this: Is his diagonal number the first number? No because its first digit is different. Is it the second? No, because the second digit is different. Likewise for all infinity, it differs from all the numbers on the list.
All you've done is fallen for his trick. His list is meaningless. Every single number he claims to have excluded from his list is a valid rational number. At no point does he ever create a number that isn't already a valid rational number, nor could he ever hope to do so. So he hasn't accomplished anything other than fooling the entire mathematical community into believing that he has proven something when in fact he hasn't proven anything at all.
mgb wrote:
His list cannot be a complete list. His list is necessarily square. But there is no such thing as numerical system that can express all possible numbers in a square list.
What has 'square' to do with it.
The fact that all possible numerical symbols must necessarily be rectangular. Period.

The mere fact that you keep asking this question indicates to me that you don't understand the nature of numerical symbolism.
mgb wrote:
So Cantor's list cannot be a complete list of all possible numbers.
That is not the point. The point is that if his list has cardinality Aleph Null it can't be a complete list. It doesn't have to be complete, it has to be infinite. There's a difference.
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.

Show me a rational number that cannot be written in decimal.
mgb wrote:
Moreover, you can also see this another way entirely.

Just stop Cantor at any point in time and look at the number he claims to have constructed that is not on his list. Ask yourself, if that number exists. Of course it exists. In fact, not only does it exist but it's also necessarily a rational number.
The point is it does not exist at the point where he is stopped or before that point. But Cantor does not stop so it does not exist anywhere.
That would only be true if a complete list of decimal numbers could be square. But it can't be square, so Cantor's method fails. He can't say that just because a number isn't on his list that it doesn't exist on an actual rectangular list. His list is ALWAYS necessarily square. Period. That can never change.

But we already know that no numerical representation of numbers can be square. Therefore Cantor's square list is meaningless.
mgb wrote:
So at no point does Cantor ever create anything more than a rational number. So while he claims to have created some magical new number he hasn't.
At no point. But we are not talking about finite points we are talking about inifinity. That's the difference.
But he hasn't shown any credible process that can be carried o ut to infinity. No matter how far he carries his process out his list will ALWAYS be square, necessarily so because of how he's constructing it.

So it's an invalid proof.

That doesn't mean that the conclusion he ultimately reaches must be false. It could be a true conclusion. But that would just be a lucky coincidence because he's proof fails to prove this.
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Post #136

Post by Divine Insight »

mgb wrote: No I can't. I await your answer. I presume you are going to say because it doesn't exist. But what does it mean for a number to exist? As far as I can see it is enough for them to exist abstractly.
Abstract concept that you don't understand are meaningless concept.

Every irrational quantitative relationship (note I am not calling them numbers) is the result of a self-referenced quantitative situation. That's what causes them to exist.

But mathematicians don't even know this. Why not? Because they made the grave error of just defining them as "abstract numbers" and stopped thinking about what they might actually be.

So mathematicians don't even know that every irrational relationship is the result of a self-referenced situation. And they will never discover this truth now because they have already embrace the absurd idea that irrational "numbers" are just abstract concepts that they simply cannot understand.

So they have chosen to go down a blind dead-end alley from which there is no return.

They'll never discover the truth of irrational relationships now because they have already accepted the concept of abstract unexplained mysterious irrational 'numbers'.

And they even treat them as numbers and will continue to do so for who knows how long?

In fact, if anyone like myself comes along and tries to point out their folly all they will do is call me a kook and continue on their merry way believing in abstract unexplained mysterious irrational numbers that they cannot explain.
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Post #137

Post by micatala »

Divine Insight wrote:
micatala wrote: How about this. Define precisely what you mean by 'complete.'

Tell me what the decimal representation of 1/3 is.
That should tell you right there that decimal representations are bogus to begin with. They can't even describe an obvious finite number in a finite way. So why are we even using decimal expansions to try to prove anything about numbers?

According to a decimal expansions 1/3 is a finite rational number that cannot even be expressed as a finite decimal expansion. So this system of notation is already exposing it's innate flaws.
micatala wrote: What is the decimal representation of the square root of 2?
The square root of 2 is an irrational relationship between two quantities that should have never been recognized as a "number" in the first place because it obviously isn't.

Mathematicians have made far more mistakes than just accepting Cantor's diagonal argument. But ironically they can't even understand why Cantor's diagonal argument fails, so I doubt they are up to the challenge of understanding many of the other errors they have made.
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. Your comments on the square root of 2 would be on par with a 5th century B.C. critic of Pythagoras.

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.

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.

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Post #138

Post by mgb »

I think I see what you are saying now. You seem to be saying that the list is growing too fast for Cantor's diagonal to catch up;

0.\
0.2 \
0.3 \

This is not a problem. Cantor's list need only contain irrationals with infinite decimal expansions. In this way the diagonal will always be in step.
All you've done is fallen for his trick. His list is meaningless. Every single number he claims to have excluded from his list is a valid rational number. At no point does he ever create a number that isn't already a valid rational number, nor could he ever hope to do so.
The number he creates is an infinite decimal expansion and is irrational. It has no repeating pattern because the numbers that generate it have no repeating pattern.
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?
That would only be true if a complete list of decimal numbers could be square. But it can't be square, so Cantor's method fails. He can't say that just because a number isn't on his list that it doesn't exist on an actual rectangular list. His list is ALWAYS necessarily square. Period. That can never change.
There is no need to have finite numbers on the list. No finite decimal expansion can be the same as his diagonal number because the diagonal number has an infinite, non repeating decimal expansion.
But he hasn't shown any credible process that can be carried o ut to infinity. No matter how far he carries his process out his list will ALWAYS be square, necessarily so because of how he's constructing it.
It doesn't matter if it is square. He can leave out all rational numbers if he wants to. He only needs to show that the irrationals are of a higher cardinality and the rationals. For this a square list of irrationals is fine.
So mathematicians don't even know that every irrational relationship is the result of a self-referenced situation. And they will never discover this truth now because they have already embrace the absurd idea that irrational "numbers" are just abstract concepts that they simply cannot understand.
I have no problem in accepting mathematics as and abstract tautology (if it is). All that matters is that it is logical.

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Post #139

Post by micatala »

mgb wrote:
Divine Insight wrote:In fact, here's a question for you,...

What causes an irrational number to be irrational?

In other words, why is the square root of 2 irrational?

And I don't just mean because it cannot be represented by a ratio of natural numbers, we already know that. I'm asking you to tell me why it cannot be represented by the ratio of natural numbers?

Can you answer me that?




No I can't. I await your answer. I presume you are going to say because it doesn't exist. But what does it mean for a number to exist? As far as I can see it is enough for them to exist abstractly.
I will handle this one.

A rational number is defined as one which is equal to the ratio of two integers, say p/q. 1/3 is rational for example.

Could the square root of two be rational? Well, if it were, it would be equal to a ratio like p/q where p and q are integers. Now, integers all have a finite number of prime factors so we could, if necessary, divide out the common prime factors between p and q. For example, 14/10 could be reduced to 7/5.

So, it is sufficient to consider the case where p and q have no common prime factors.

So, suppose (root 2)=p/q and that p and q have no common prime factors.
It follows that 2 = (p^2)/(q^2) which implies 2q^2=p^2. We have 2 is a factor of the left hand side of this last equation which means 2 must be a factor of the right hand side, namely p^2. But, an odd number squared cannot have a factor of 2. Therefore, p is even. This means (definition of even) p=2k for some integer k.

Now substitute 2k for p in the equation 2q^2=p^2. We get

2q^2=(2k)^2=4k^2. This implies q^2=2k^2.

But, as before, this can only be true if q is even. Thus, we have the logical result that both p and q are even, in other words, have a common factor of 2.

But we already had reduced to the case where p and q have no common factors, so we have a contradiction. If root 2 is rational, then two contradictory statements both have to be true. Namely, root 2 can be expressed as a fraction in lowest terms, and root 2 cannot be expressed as a fraction in lowest terms

Thus, the original assumption that root 2 was rational was incorrect. Therefore, root 2 is irrational.

Legend has it that a follower of Pythagoras discovered the irrationality of root 2 and the great man and his followers were so upset that they through the unfortunate individual into the sea to prevent word from getting out.
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Post #140

Post by micatala »

[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.
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