Why Cantor's Diagonalization Proof is Flawed.

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Divine Insight
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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 #121

Post by micatala »

mgb wrote:
DivineInsight wrote:I can't believe you just said this.
How are the natural numbers not infinite?
You may be confusing 'complete' with 'infinite'. Just because Aleph Null is infinite does not mean it is complete. The list of naturals is infinite but not complete because it does not contain fractions.

Cantor's argument is that if we assume a list of reals has cardinality Aleph Null that list would be infinite but not a complete list of reals. This is the core of his argument and he shows it is correct by producing a number not on the infinite list. Therefore the cardinality of the reals is greater than Aleph Null.
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.
Nothing needs to be completed in actual terms. Abstractly 0.333... is infinite and complete. That is all that is needed.
micatala wrote:I am not following you. No integer has an infinite number of digits. You can create an infinite list of integers, for example

1, 10, 100, 1000, . . .

but every integer in the list will have a finite number of digits.
But if you can have an infinite decimal expansion you can have infinite integers! Just imagine Pi without the decimal point. (I'm just speculating here...)
I think a lot of Divine Insight's confusion is an imprecise and probably inconsistent understanding of the term 'complete.'

In terms of Cantor's proof, Cantor is proceeding by reductio ad absurdum, proof by contradiction. For purposes of contradiction, he assumes (as you state here) that the cardinality of the reals is aleph null. By definition, this means there is a one-to-one correspondence between the set of real numbers and the natural numbers. Cantor represents this by what we might call a 'complete list' of the real numbers. All 'complete' means here is that every real number is in the list, and the list thus represents that posited one-to-one matching.

The decimal representation of the numbers IN the list is simply for convenience. One could conceivably make the 'assumption for purposes of contradiction' and never even write down a decimal representation.

However, 'complete' also has another more technical connotation which mgb is hinting at here I think. A compete set is a set in a metric space satisfying the condition that every so-called Cauchy sequence (essentially one that converges) converges to a number that is in the same set.

The set S of rational numbers with finite decimal expansions, for example, is NOT complete. Consider the examples I had a page or two back. Here are two Cauchy sequences within the set S.

a_n: 0.3, 0.33, 0.333, . . .
b_n: 0.4, 0.34, 0.334, . . .

These both converge to 1/3, which is rational, but has no finite decimal expansion, so is not in S. In an intuitive sense, S has a 'hole' at 1/3. There are elements of S as close as one would like to 1/3 but 1/3 is not in S.

Discrete sets like the natural numbers are also not compete, but that is somewhat of a vacuous statement as there are no Cauchy sequences in the natural numbers. Cauchy essentially means that the terms of the sequence 'get closer and closer to each other' the further down the list you go.

The rational numbers are not complete since there is a sequence of rational numbers that converges to the square root of 2, and root 2 is not rational.

The real numbers ARE compete in this more technical sense.

However, this type of completeness is really a separate issue from the argument that the real numbers are uncountable, that is, of a higher cardinality than the natural numbers.
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Post #122

Post by mgb »

micatala wrote:In terms of Cantor's proof, Cantor is proceeding by reductio ad absurdum, proof by contradiction. For purposes of contradiction, he assumes (as you state here) that the cardinality of the reals is aleph null. By definition, this means there is a one-to-one correspondence between the set of real numbers and the natural numbers. Cantor represents this by what we might call a 'complete list' of the real numbers. All 'complete' means here is that every real number is in the list, and the list thus represents that posited one-to-one matching.
Yes. And the converse is that if the list is complete (and even contains the diagonal number) its cardinality must be greater than Aleph Null. But this is a brain warping way to look at it!

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Post #123

Post by Divine Insight »

micatala wrote: As noted before, the mathematics of limits invented by Newton got us into space and to the moon, among other things.
Forget about Newton. Our modern Calculus is defined by the epsilon-delta definition refined by Karl Weierstrass.

I'm totally on-board with Calculus and the definition of the limit. I'm not denying calculus at all.

All I'm saying is that if you think that the existence of a calculus limit demands that the limit must actually exist in the real world, then it's you who doesn't understand the Calculus limit.

My position is in total harmony with all of Calculus.
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Re: Why Cantor's Diagonalization Proof is Flawed.

Post #124

Post by Divine Insight »

micatala wrote: You continue to confuse the actual value of a number with its decimal representation. My list is the same four numbers as your previous 2 x 4 representation. If you define 'complete' to be all possible numbers that can be represented by two digit binary integers, then my list, being the same exact set of numbers, is also complete.

Changing the representation does not change the set.
As long as your promise to NEVER place anything after your decimal points but zeros then I would agree. However, if you are going to allow any numerals other than zero to be placed in those positions then you are dead wrong and you'll need to add additional rows to make your list complete.

So your trick doesn't work.
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Post #125

Post by Divine Insight »

micatala wrote: We have but you have stubbornly refused to acknowledge it and correct the error of your own thinking.
You have not shown where I've been wrong about anything.

micatala wrote:
In fact you yourself have demonstrated you lack of understanding in your very next post when you propose that by just adding a few more zeros to an existing list will make it square.
I did make it square, following your own criterion. I simply represented the exact same set with more digits, just like I could represent 1/4 as 0.25 or 0.2500 or 50/200.

Surely you would not deny that the finite decimal 0.25 is equal to the finite decimal 0.2500 would you?
As long as you NEVER change those additional zeros to anything other than zeros I will agree.

But that's not going to help you in the case of Cantor's proof.
micatala wrote:
You can't just add more digits without taking into consideration how many more rows you will also need to add to accommodate a completed list. But that's exactly what you did.
Yes, I can. You have given no one any reason to think otherwise. 0.25=0.250=0.2500.
You aren't even addressing the original problem at all.

Apparently you haven't yet understood the problem.

We are talking about making complete lists of numbers. We aren't talking about adding zeros to a single numerical representation and recognizing that this doesn't change the value.

For crying out loud, do you really think I that stupid?

You aren't even understanding the original problem at all.
micatala wrote:
So you clearly do not even understand the problem.
You are incorrect. It is the exact opposite. You continue to make up things that 'need to be' a certain way without basis in logic, you confuse representations of numbers with the actual sets of numbers, and you deny the basic mathematics of infinite decimals.
Absolute hogwash.

We are talking about making complete lists. Period.

Something you seem to be totally ignoring.
micatala wrote: Again, do you deny that 0.25=0.250-0.2500?
Of course not. What do you think I am? An idiot?

That question has absolutely nothing at all do to with whether a complete list of numerals can be made to be square.

Apparently you've never understood the original problem.

And still don't understand it yet.
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Post #126

Post by Divine Insight »

micatala wrote: I think a lot of Divine Insight's confusion is an imprecise and probably inconsistent understanding of the term 'complete.'
There is no confusion here.

In fact, I'm not even remotely arguing against the conclusion that Cantor arrives at.

I have other reasons why the so-called real numbers cannot be placed into a one-to-one correspondence with the natural numbers. So I'm not even contesting that conclusion at all.

All I'm doing is pointing out the flaw in Cantor's diagonal argument.

Unless you can show how a complete list of numerals can be made square, then my argument stands and Cantor's proof fails.
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Post #127

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Divine Insight wrote:
micatala wrote: I think a lot of Divine Insight's confusion is an imprecise and probably inconsistent understanding of the term 'complete.'
There is no confusion here.

In fact, I'm not even remotely arguing against the conclusion that Cantor arrives at.

I have other reasons why the so-called real numbers cannot be placed into a one-to-one correspondence with the natural numbers. So I'm not even contesting that conclusion at all.

All I'm doing is pointing out the flaw in Cantor's diagonal argument.

Unless you can show how a complete list of numerals can be made square, then my argument stands and Cantor's proof fails.
How about this. Define precisely what you mean by 'complete.'

Tell me what the decimal representation of 1/3 is.

What is the decimal representation of the square root of 2?
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Post #128

Post by mgb »

Divine Insight wrote:
micatala wrote: I think a lot of Divine Insight's confusion is an imprecise and probably inconsistent understanding of the term 'complete.'
There is no confusion here.

In fact, I'm not even remotely arguing against the conclusion that Cantor arrives at.

I have other reasons why the so-called real numbers cannot be placed into a one-to-one correspondence with the natural numbers. So I'm not even contesting that conclusion at all.

All I'm doing is pointing out the flaw in Cantor's diagonal argument.

Unless you can show how a complete list of numerals can be made square, then my argument stands and Cantor's proof fails.
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.
DivineInsight wrote: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 diagonal moves as fast as the list grows:

24
37

the diagonal is 27 which is 2 digits.

245
371
845

now the diag. is 275, so I grows as fast as the list grows (if the decimal expansions are infinite there is no problem).
DivineInsight wrote:Yet Cantor claims to be creating a "Completed List" here.
No he doesn't. He says that if its cardinal number is Aleph Null it can't be complete. He shows it is not complete by creating a diagonal number not on the list (if its length is Aleph Null).
He then concludes that its cardinality must be greater than Aleph Null. Easy. Where's the problem?

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Post #129

Post by Divine Insight »

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.
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Post #130

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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.

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