Why Cantor's Diagonalization Proof is Flawed.

Discuss Physics, Astronomy, Cosmology, Biology, Chemistry, Archaeology, Geology, Math, Technology

Moderator: Moderators

Post Reply
User avatar
Divine Insight
Savant
Posts: 18070
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 19 times

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.
[center]Image
Spiritual Growth - A person's continual assessment
of how well they believe they are doing
relative to what they believe a personal God expects of them.
[/center]

keithprosser3

Post #61

Post by keithprosser3 »

If it can't be done in a finite situation then how on God's Green Earth could it be extended to infinity?
Oh, well... does anyone have a use for a very,very large piece of paper covered in numbers? I have one I won't be needing.

User avatar
micatala
Site Supporter
Posts: 8338
Joined: Sun Feb 27, 2005 2:04 pm

Post #62

Post by micatala »

Divine Insight wrote:
keithprosser3 wrote: 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? :-k

Firstly, your last statement is fallacious and so ALL of your analogies, and you were the one who started this, with finite lists are irrelevant.


Consider that, in a finite situation, a set cannot be put into one-to-one correspondence with a proper subset of itself. The one-digit integers cannot be put into correspondence with {1, 2, 3, 4, 5} for example.

However, with infinite sets this is not the case. the function f(n) = 2n is a one-to-one correspondence between the natural numbers and the proper subset of even natural numbers.

So, we have yet another reason your arguments completely fall apart. All your talk about finite rectangular arrays of numbers are entirely irrelevant to Cantor's proof. Thus, you have not shown Cantor's proof is flawed in any way.



I again ask the question which you keep ignoring.


Can you show any actual logical (not subjective) absurdity regarding the correspondence between real numbers between 0 and 1 and infinite decimal expansions of the form 0.a1 a2 a3 a4 . . . ?


If you are going to actually address Cantor's proof, instead of these fallacious and ill-defined straw men you keep erecting, you need to address this key point for one.
" . . . the line separating good and evil passes, not through states, nor between classes, nor between political parties either, but right through every human heart . . . ." Alexander Solzhenitsyn

User avatar
Divine Insight
Savant
Posts: 18070
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 19 times

Post #63

Post by Divine Insight »

micatala wrote: So, we have yet another reason your arguments completely fall apart. All your talk about finite rectangular arrays of numbers are entirely irrelevant to Cantor's proof. Thus, you have not shown Cantor's proof is flawed in any way.
I absolute have. I have shown that is has absolutely no logical credibility at all.

If a process can't even begin to make sense in a finite case, then what justification does can't have for imagining to "extend" that process to infinity?

He can't even show that his process is a valid process to begin with.
micatala wrote: I again ask the question which you keep ignoring.


Can you show any actual logical (not subjective) absurdity regarding the correspondence between real numbers between 0 and 1 and infinite decimal expansions of the form 0.a1 a2 a3 a4 . . . ?


If you are going to actually address Cantor's proof, instead of these fallacious and ill-defined straw men you keep erecting, you need to address this key point for one.
I am not erecting any straw men, that is a false accusation on your behalf. It's also a rather insulting accusation since you are basically accusing me of purposefully construction fallacious arguments. You are basically accusing me of being dishonest. I resent your insinuations here.

I have not given any ill-defined strawman arguments at all. I have clearly defined every concept I am using in enough detail that even a grade school student should be able to understand them.

I have proven beyond a reasonable doubt that any "Complete List" of numbers written out using numerical notation must necessarily be far taller than it is wide and that no diagonal line that crosses off consecutive digits with every new row on such a list could ever hope to descend through the list at a steep enough slope to achieve Cantor's goal.

This observation alone totally destroys Cantor's diagonalization argument.

So I'm actually already done at this point. I've already proven that Cantor's diagonalization method is bogus and horribly flawed in terms of logic. I've blown his diagonalization method clean out of the water at this point.

No further explanation should even be required at this point.

However, I would like to put to you the following question:

At what point does Cantor's method actually produce a number that is not a legitimate Rational Number?

At every step of his process, the "NEW" decimal number that he has created is a legitimate Rational Number.

Just stop his process at any point you so desire, it doesn't matter when you chose to stop his process because this will always be the result.

What will you have at that point in his process?

Well, since you've stopped you will have a truncated decimal. Every truncated decimal number is a legitimate Rational Number. Cantor's method, NEVER at any time actually produces anything other than a legitimate Rational Number at any given step of his process. He never creates a decimal number that isn't already a valid rational number.

So my question is this, "At what point does Cantor's method actually produce a number that is not a legitimate Rational Number?"

When does he claim to do this? At infinity? At the infinite^th step of his process?

What "magic" has occurred at that step that had never occurred before?

His process could only make any sense at all if, and only if, it could be completed. Because his "magic" only works at the infinite^th step of his process.

Any time prior to this he is just standing there holding a valid Rational Number.

It's a bogus demonstration. It only demonstrates that Cantor himself wasn't thinking clearly.

Not only did he not take into account the fact that a diagonal line can't even descend a completed numerical list at a fast enough rate, but his process also assumes (and demands) that he must be able to complete this infinite process in order to have his "magic" occur.

Because at any point prior to his magical "infinite^th" final step he always has produced nothing more than a valid Rational Number.

His process can only work in his dreams. It can only be meaningful for someone who can dream of actually completing an infinite process. And even that makes no sense, because what would that mean to have "completed" the process? If he actually stopped he'd be embarrassingly standing there holding a valid Rational Number in his hands.

His argument is fallacious. It's is not built upon sound logic at all. It's a totally bogus argument. It fails for two main reasons:

1. He could never construct a "completed List" of numerals by constructing that list using a diagonal line that crosses off each consecutive digit in each consecutive row. (This is the obvious finite truth that totally blows his method clean out of the water)

2. His method would never produce this mystical number that he claims to be generating anyway unless his method could somehow be "completed".


So his very conclusion that he could generate such a number is entirely dependent upon accepting his notion of "Completed Infinities". Which itself is a contradiction in terms. If the process were to ever actually be completed (i.e. terminated) it would only produce a valid Rational Number at that point anyway.

And if it is never completed, then the proof falls apart anyway because it never succeeds in actually producing this magical number that supposedly can't be put into a one-to-one correspondence with the set of Natural Numbers.

It's a totally bogus proof.

That fact that it has existed in math textbooks for the past couple of centuries is a testament to the sloppiness that has become part and parcel of Modern Day Mathematics.

It's a totally bogus argument. And Henri Poincare was indeed right, bless his soul.
[center]Image
Spiritual Growth - A person's continual assessment
of how well they believe they are doing
relative to what they believe a personal God expects of them.
[/center]

olavisjo
Site Supporter
Posts: 2749
Joined: Tue Jan 01, 2008 8:20 pm
Location: Pittsburgh, PA

Post #64

Post by olavisjo »

.
lao tzu wrote: The above is flawed in that it includes the incorrect assumption that all numbers are listable.
All numbers are listable.

Here is a list of all real numbers R; where 0.0 < R <= 1.0
And they are in a one-to-one correspondence with the natural numbers.

1 - 0.1
2 - 0.2
3 - 0.3
...
9 - 0.9
10 - 0.01
11 - 0.11
12 - 0.21
...
99 - 0.99
100 - 0.001
101 - 0.101
102 - 0.201
...


Similarly, the entire set of real numbers can be placed in a one-to-one correspondence with the natural numbers.

http://debatingchristianity.com/forum/v ... 630#607630
"I believe in no religion. There is absolutely no proof for any of them, and from a philosophical standpoint Christianity is not even the best. All religions, that is, all mythologies to give them their proper name, are merely man’s own invention..."

C.S. Lewis

lao tzu
Apprentice
Posts: 106
Joined: Mon Dec 03, 2007 1:04 pm
Location: Everglades

Post #65

Post by lao tzu »

olavisjo wrote: .
lao tzu wrote: The above is flawed in that it includes the incorrect assumption that all numbers are listable.
All numbers are listable.

Here is a list of all real numbers R; where 0.0 < R <= 1.0
And they are in a one-to-one correspondence with the natural numbers.

1 - 0.1
2 - 0.2
3 - 0.3
...
9 - 0.9
10 - 0.01
11 - 0.11
12 - 0.21
...
99 - 0.99
100 - 0.001
101 - 0.101
102 - 0.201
...


Similarly, the entire set of real numbers can be placed in a one-to-one correspondence with the natural numbers.

http://debatingchristianity.com/forum/v ... 630#607630
Dear olavisjo,

Every number on your list ends in terminal zeros, and hence cannot be complete. Your list excludes not only all transcendentals, not only all the algebraics, but also all rationals with denominators whose prime factors include anything other than 2 or 5.

If you wish to disprove this blanket statement, which goes far beyond a mere disproof of your thesis that all real numbers are listable, you need only name the integer corresponding to the fraction 1/3.

Regards, J

ETA: Looking over the thread you link, I discover this omission has already been pointed out to you, and rather than answering the objection, you have instead chosen to shift the broken argument to another venue, apparently in the belief that this time you'll be able to get away with it.

This is beyond bad form, and indeed indicative of an argument presented in bad faith. We are done here. You will waste no more of my time.
There is no lao tzu.

olavisjo
Site Supporter
Posts: 2749
Joined: Tue Jan 01, 2008 8:20 pm
Location: Pittsburgh, PA

Post #66

Post by olavisjo »

.
keithprosser3 wrote: Well, first off Cantor was dealing with real numbers not integers. Does that matter? Yes. For convenience only it is usual to consider only (real) numbers between 0 and 1 so if we want 'squares of numbers' it's easy:

1) 0.0001 4 digits
2) 0.2345
3) 0.4534
4) 0.4567

which becomes
1) 0.00010 if you want 5 digits, just right pad with zeros.
2) 0.23450
3) 0.45340
4) 0.45670
5) 0.67543

The numbers ahead of the ) are just to indicate clearly the position or index of each real in the list of reals.

To make the square as wide as it is high, just pad with zeros on the right as required, as that doesn't change the value of the number.
Here you say "just right pad with zeros" as if that is a trivial solution to the problem that DI posed. I will show you why that is not valid by doing this problem in binary, if Cantor's theorem is valid in base ten it should also be valid in binary. And it is this "just pad with zeroes" that is the Achilles' heel (literally) in Cantor's argument.

First I will list all the binary numbers n: where 0.0 <= n <= 1.0

Code: Select all

     0&#41;        0     0.0000000... 
     1&#41;        1     0.1000000... 
     2&#41;       10     0.0100000...
     3&#41;       11     0.1100000...
     4&#41;      100     0.0010000...
     5&#41;      101     0.1010000...
     6&#41;      110     0.0110000...
     7&#41;      111     0.1110000...
     8&#41;     1000     0.0001000...
   ...&#41;      ...     ...
...999&#41;   ...111     0.1111111... 
We can know the value of any digit f(x,y) where x is the number of digits accross and y is the number of rows down.

For example, the third digit in the seventh row would be 1 or f(3,7) = 1

Where f(x,y) = if ( (y % 2x)/2x < 1/2) then 0; else 1



Now we will look at the diagonals.
The first diagonal is 0.0000000... we know that the last digits will always be zero because our formula f(x,y) will return zero when 2x > 2y
Then we will change the digits from 0.0000000... to 0.1111111...
But this number is on the list.

The second diagonal is 0.1100000... again we will flip the digits and get 0.0011111...
But, 0.0011111... is equal to 0.0100000... (if you don't believe me look up Zeno's paradox) which is on the list.

The third diagonal is 0.0110000... again we will flip the digits and get 0.1001111...
But, 0.1001111... is equal to 0.1010000... which is on the list.

The fourth diagonal is 0.1010000... again we will flip the digits and get 0.0101111...
But, 0.0101111... is equal to 0.0110000... which is on the list.

And so on and on.

So, if Cantor's proof does not work in all numeric bases, then it can be said that it is invalid.
"I believe in no religion. There is absolutely no proof for any of them, and from a philosophical standpoint Christianity is not even the best. All religions, that is, all mythologies to give them their proper name, are merely man’s own invention..."

C.S. Lewis

User avatar
Divine Insight
Savant
Posts: 18070
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 19 times

Post #67

Post by Divine Insight »

keithprosser3 wrote: Well, first off Cantor was dealing with real numbers not integers. Does that matter? Yes. For convenience only it is usual to consider only (real) numbers between 0 and 1 so if we want 'squares of numbers' it's easy:

1) 0.0001 4 digits
2) 0.2345
3) 0.4534
4) 0.4567

which becomes
1) 0.00010 if you want 5 digits, just right pad with zeros.
2) 0.23450
3) 0.45340
4) 0.45670
5) 0.67543

The numbers ahead of the ) are just to indicate clearly the position or index of each real in the list of reals.

To make the square as wide as it is high, just pad with zeros on the right as required, as that doesn't change the value of the number.
Padding zeroes to the right in decimal notation doesn't help. In fact, this notatoin actually serves to obscure Cantor's folly, because it appears to justify his assumption that a list of real decimal numbers could be "made square" simply by padding to the right.

But you have the same problem in any case.

Let's say that we are going to list the rational numbers using the following method.

1: 0.1
2: 0.2
3: 0.3
4: 0.4
5: 0.5
6: 0.6
7: 0.7
8: 0.8
9: 0.9
10: 0.10
11: 0.11
12: 0.12
13: 0.13

And so on.

We'd have doubles (Like the first row and the tenth row), but that's ok. In fact, the 10th row is already starting to pad zeroes.

But hopefully you can see the problem already. It's going to be the very same problem we had with a list of Integers. You can't run a diagonal line down this list fast enough (at a steep enough slope) to cover every possible numerical representation of a decimal number.

In words, take the first two rows and draw you diagonal line. You the first 1 and replace that with say a 2. Then when you get to row 2 you must "pad" in a new zero. So you then change that zero to say a 1. So your new number is:

0.21

That's the new number that you have created on this list. But you are only working at ROW #2.

Now let's look down at row number 21 on the actual "completed list". What do we see? We we see the following:

21: 0.21

It's already on the list. It's just way down the list at row number 21.

Now think about this.

This will ALWAYS BE TRUE.

This will always be true at every step of Cantor's diagonalization process.

He will always be way behind where he needs to be to "catch up" to any actual real completed list.

Let's say after 10 steps he comes up with the real decimal number 0.2111111111

Has he created a new number that isn't on a completed list?

No he has not. In fact, we can even show precisely where that number will be on our ordered list.

It will be in the 2111111111th row.

2111111111: 0.2111111111

But Cantor is only working back on row 10.

He could never hope to keep up with a "completed list" of decimal numbers.

He get's exponentially further behind with every new number he creates.

So padding zeroes to the right doesn't help his problem.

Moreover, look again at my original list:

1: 0.1
2: 0.2
3: 0.3
4: 0.4
5: 0.5
6: 0.6
7: 0.7
8: 0.8
9: 0.9
10: 0.10
11: 0.11
12: 0.12
13: 0.13

We see that the Integers 1 and 10 both map to the same decimal number (precisely because padding to the right with zeroes doesn't produce a new decimal number)

This will also be true for the integers 100, 1000, 10000, 1000000, etc.

Therefore we can stick Pi in at row 10. We can stick the square root of 2 in at row 100. We can't stick e in at row 1000 and so on. (lop off the whole number start of these numbers of course because we're only working from 0 to 1.)

In other words, we even have extra room on our list to accommodate irrational numbers.

Although I should add at this point that the first digits of Pi, the square root of 2, e, and all other irrational numbers will be on this list infinitely often anyway.

If you truncate any of these irrational numbers at any time you will have a valid rational number, because all rational numbers will be on this list.

In other words, here's Pi taken out to some arbitrary decimal place.

3.1415926535897932384626433832795

Lope off the 3 and we have 0.1415926535897932384626433832795

Will this number be on our list? Yes, we can even point to what row it will be in:

1415926535897932384626433832795: 0.1415926535897932384626433832795

And every occurrence of Pi (with the 3 lopped off) taken out to as many decimal places as you wish will be on this list.

In theory you can take this out to infinitely many decimal places.

Why?

Well isn't this supposed to be an INFINITE LIST?

That's another demand that Cantor is putting onto this whole "proof".

So padding with zeroes to the right doesn't solve the problem. It only serves to hide the folly of the so-called proof.
[center]Image
Spiritual Growth - A person's continual assessment
of how well they believe they are doing
relative to what they believe a personal God expects of them.
[/center]

keithprosser3

Post #68

Post by keithprosser3 »

I'm posting so you know I am not ignoring you DI. But I would only be repeating myself. We'll just have to agree to disagree or something as neither of us could ever convince the other. There's always lao tzu to argue with!

btw - I'll be posting much less on the internet from from now on - I need to spend more time on other stuff.

User avatar
Divine Insight
Savant
Posts: 18070
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 19 times

Post #69

Post by Divine Insight »

keithprosser3 wrote: btw - I'll be posting much less on the internet from from now on - I need to spend more time on other stuff.
That's what I need to be doing too. ;)
[center]Image
Spiritual Growth - A person's continual assessment
of how well they believe they are doing
relative to what they believe a personal God expects of them.
[/center]

olavisjo
Site Supporter
Posts: 2749
Joined: Tue Jan 01, 2008 8:20 pm
Location: Pittsburgh, PA

Post #70

Post by olavisjo »

.
Divine Insight wrote: Let's say that we are going to list the rational numbers using the following method.

1: 0.1
2: 0.2
3: 0.3
4: 0.4
5: 0.5
6: 0.6
7: 0.7
8: 0.8
9: 0.9
10: 0.10
11: 0.11
12: 0.12
13: 0.13

And so on.

We'd have doubles (Like the first row and the tenth row), but that's ok. In fact, the 10th row is already starting to pad zeroes.
It is not okay to have doubles. It means that something is missing, in this case 0.01 is missing from your list.

This is why I inverted the digits to form my list. So, 10 becomes 0.01, and nothing will be missing.
Divine Insight wrote: Therefore we can stick Pi in at row 10. We can stick the square root of 2 in at row 100.
You can't stick these numbers just anywhere, or you will lose your one to one correspondence. These numbers will be on the list, but they will be an infinite distance down the list.
People will object by saying 'if it is an infinite distance down the list, you will never get there'. This misunderstanding is based on the false assumption that the list is going to infinity, it is not going there, but rather it is already there.
For example, 'the number 0.999... is going to get very close to 1.0' is false, as it is not going to 1.0, but rather as soon as you put the three dots behind the 9's, it is 1.0, there is no going.
"I believe in no religion. There is absolutely no proof for any of them, and from a philosophical standpoint Christianity is not even the best. All religions, that is, all mythologies to give them their proper name, are merely man’s own invention..."

C.S. Lewis

Post Reply