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: 18080
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 15 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]

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

Post #111

Post by micatala »

Divine Insight wrote:
mgb wrote: A square is a finite entity. You need to accept that finite arithmetic does not apply to infinities.

Apparently I might be the only person on planet earth who has caught this error.
I hate to be unkind, but you might consider the possibility that the reason your are the 'only person on the planet' making the argument you are making is that you are the one who is wrong.

The entire premise of your argument is to assume something about Cantor's proof that is not part of the proof. You are engaging a straw man with your whole 'square lists of numbers' argument.

And the fact that you refuse to accept the logical soundness of 0.333 . . . =1/3 is further evidence that you simply do not understand the arguments you are trying to refute. Again, I hate to be unkind, but after 11 pages of this there is not much else to say. You are like Bishop Berkeley writing a whole essay railing against Newtonian Calculus without really understanding what Calculus was. To be fair, Newton's work writing was difficult to engage with and the arguments provided by Cauchy and others which provided the logical soundness of Calculus did not yet exist in Berkeley's day.

But those arguments have been put to bed over 150 years ago now. And Newton's Calculus was sound enough to put satellites into orbit and a man on the moon. According to you, we should reject it. I confess myself perplexed at what appears to me to be a continued commitment to a profound misunderstanding.
" . . . 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
micatala
Site Supporter
Posts: 8338
Joined: Sun Feb 27, 2005 2:04 pm

Re: Why Cantor's Diagonalization Proof is Flawed.

Post #112

Post by micatala »

Divine Insight wrote: [Replying to post 97 by benchwarmer]


I don't see your explanation as being a valid representation of what Cantor is doing.

For one thing, if we accept your explanation then we can apply it to anything, including the natural numbers. So what could it possibly be telling us about the natural numbers?

As far as I see it, Cantor's list must be a "complete" list of all possible numbers. Otherwise saying that he can create numbers that aren't on his list is meaningless.

Take my original binary example.

Here is the complete list of all possible numbers that can be expressed by using binary numerals that are only two digits wide.

00
01
10
11
00.00
01.00
10.00
11.00

Your list is now square. It is no more relevant to Cantor's argument than it was before, but it is square.
" . . . 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: 18080
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 15 times

Post #113

Post by Divine Insight »

micatala wrote: Well, I am grateful for the clarification. I would point out that if you refuse to acknowledge the logical soundness of infinite repeating decimal representations, you are rejecting mathematics that predates Cantor and is taught in nearly every high school in the U.S.

A "Calculus Limit" is a quantity. Calculus limits that exist are numbers, just like numbers defined in any other way.
This is absolutely false.

A Calculus limit is not a valid quantity. It's not even remotely defined as such. A Calculus limit is simply a statement of what a particular mathematical expression or operation is heading toward. It's a trend, not an absolute quantity.

In fact, this is another place where mathematicians grossly dropped the ball. Ask almost any mathematician for the solution to Zeno's paradox and they will point to Calculus. But nothing could be further from the truth. Calculus not only doesn't resolve Zeno's paradox, but instead it actually confirms that Zeno was correct.

In fact, Quantum Mechanics had already shown that Zeno was indeed correct and that our world is indeed quantized and not a continuum. So Zeno was right all along, and mathematicians drop the ball again.

To say that 1/2=0.5=0.50000 . . . = 0.49999 . . . is false, and totally incorrect UNLESS you end that statement in the following way,...

1/2=0.5=0.50000 . . . = 0.49999 . . . in the Calculus limit.

Then you have spoken a truth, but that isn't the same as claiming that 1/2=0.5=0.50000 . . . = 0.49999 . . . in any absolute sense.

Mathematicians not have not only lost their way, but apparently they are teaching all their students to perpetuate these same errors in logic.
[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]

User avatar
Divine Insight
Savant
Posts: 18080
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 15 times

Re: Why Cantor's Diagonalization Proof is Flawed.

Post #114

Post by Divine Insight »

micatala wrote: 00.00
01.00
10.00
11.00

Your list is now square. It is no more relevant to Cantor's argument than it was before, but it is square.
And you have just made the same error that Cantor made all over again.

You've expanded your columns to 4 places while keeping only four rows.

That's not going to work.

If you are allowing your binary numerals to be four digits long, then you need to have 16 rows below them to be a "complete" list.

So you apparently haven't even understood the problem. You seem to be thinking that you can just add as many columns as you like without taking into consideration how many rows would be required to hold a complete list of the resulting numerals.

That's the same error that Cantor made.
[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]

User avatar
Divine Insight
Savant
Posts: 18080
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 15 times

Post #115

Post by Divine Insight »

micatala wrote: I hate to be unkind, but you might consider the possibility that the reason your are the 'only person on the planet' making the argument you are making is that you are the one who is wrong.
I'll be more than happy to concede to being wrong when someone demonstrates the truth of that accusation.

Thus far no one has even been able to demonstrate an understanding of the error that Cantor made, much less address it.

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.

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.

So you clearly do not even understand the problem.
[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]

User avatar
Divine Insight
Savant
Posts: 18080
Joined: Thu Jun 28, 2012 10:59 pm
Location: Here & Now
Been thanked: 15 times

Post #116

Post by Divine Insight »

micatala wrote: But those arguments have been put to bed over 150 years ago now. And Newton's Calculus was sound enough to put satellites into orbit and a man on the moon. According to you, we should reject it. I confess myself perplexed at what appears to me to be a continued commitment to a profound misunderstanding.
By the way, nowhere did I ever reject the Calculus.

Also, when engineers use Calculus to go to the moon they never used any infinite answers. In fact, that couldn't even be done no matter what. All Calculus does is allow us to calculate trends and from that we can actually make useful finite calculations that can get us to the moon.

Trust me, no infinite quantities were required to go to the moon, for if they were we would have never been able to do it.

I don't renounce calculus, I simply understand what it is and I don't try to use it to make absurd claims about things it can't even truly address.
[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]

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

Post #117

Post by micatala »

Divine Insight wrote:
micatala wrote: Well, I am grateful for the clarification. I would point out that if you refuse to acknowledge the logical soundness of infinite repeating decimal representations, you are rejecting mathematics that predates Cantor and is taught in nearly every high school in the U.S.

A "Calculus Limit" is a quantity. Calculus limits that exist are numbers, just like numbers defined in any other way.
This is absolutely false.

A Calculus limit is not a valid quantity. It's not even remotely defined as such. A Calculus limit is simply a statement of what a particular mathematical expression or operation is heading toward. It's a trend, not an absolute quantity.

In fact, this is another place where mathematicians grossly dropped the ball. Ask almost any mathematician for the solution to Zeno's paradox and they will point to Calculus. But nothing could be further from the truth. Calculus not only doesn't resolve Zeno's paradox, but instead it actually confirms that Zeno was correct.

In fact, Quantum Mechanics had already shown that Zeno was indeed correct and that our world is indeed quantized and not a continuum. So Zeno was right all along, and mathematicians drop the ball again.

To say that 1/2=0.5=0.50000 . . . = 0.49999 . . . is false, and totally incorrect UNLESS you end that statement in the following way,...

1/2=0.5=0.50000 . . . = 0.49999 . . . in the Calculus limit.

Then you have spoken a truth, but that isn't the same as claiming that 1/2=0.5=0.50000 . . . = 0.49999 . . . in any absolute sense.

Mathematicians not have not only lost their way, but apparently they are teaching all their students to perpetuate these same errors in logic.
You continue to engage in inconsistent and illogical arguments, and you deny mathematics at the high school level. Your arguments display a profound misunderstanding of the real number system and calculus. Arithmetic with numbers defined as limits is every bit as logically valid, if somewhat more complicated, than with integers.

As noted before, the mathematics of limits invented by Newton got us into space and to the moon, among other things.
Last edited by micatala on Mon Apr 22, 2019 9:48 am, edited 1 time in total.
" . . . 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
micatala
Site Supporter
Posts: 8338
Joined: Sun Feb 27, 2005 2:04 pm

Re: Why Cantor's Diagonalization Proof is Flawed.

Post #118

Post by micatala »

Divine Insight wrote:
micatala wrote: 00.00
01.00
10.00
11.00

Your list is now square. It is no more relevant to Cantor's argument than it was before, but it is square.
And you have just made the same error that Cantor made all over again.

You've expanded your columns to 4 places while keeping only four rows.

That's not going to work.

If you are allowing your binary numerals to be four digits long, then you need to have 16 rows below them to be a "complete" list.
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.

If you want to move the goalposts every time the representation is done in a different way, then you are engaging in a logical inconsistency.

Granted, I could define a larger set using my representation with two digits after the decimal point, but the representation does not dictated the set. The representation is simply a different way of displaying whatever set you have under consideration.

Cantor was representing real numbers using decimal expansions. This clearly requires infinitely many numbers since, containing the infinite set of integers, the real numbers are infinite.

It also requires some of the individual numbers to be infinite, like 1/3. You deny that
1/3=0.3333333. Pray tell, what is your decimal expansion for 1/3? How about for the square root of 2 or pi?

You continue to insist, without basis in fact or logial and after my pointing out a counterexample, that arguments on finite sets have to go over to infinite sets.
So you apparently haven't even understood the problem. You seem to be thinking that you can just add as many columns as you like without taking into consideration how many rows would be required to hold a complete list of the resulting numerals.
You continue to confuse representations with sets of numbers. My set of numbers is the same as yours. I think part of the problem is you have an inconsistent understanding of what you mean by 'complete.' You think 'complete' goes with the representation.



For Cantor's argument, a 'complete list' simply means one that contains all the real numbers, regardless of how they are represented. He could have represented at least some of the numbers in the list (the rational ones and the integers) with fractions. He chose the representation as a tool for the proof.

It's analogous to representing x^2-4 as (x+2)(X-2). They are the same polynomial and the same as functions, just represented differently. We could also represent this using summation notation in the abstract via Taylor series. We don't do that typically for polynomials since all the terms of degree higher than 2 would have a zero coefficient (just like 0.5 is the same as 0.5000 . . . ).
" . . . 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
micatala
Site Supporter
Posts: 8338
Joined: Sun Feb 27, 2005 2:04 pm

Post #119

Post by micatala »

Divine Insight wrote:
micatala wrote: I hate to be unkind, but you might consider the possibility that the reason your are the 'only person on the planet' making the argument you are making is that you are the one who is wrong.
I'll be more than happy to concede to being wrong when someone demonstrates the truth of that accusation.

Thus far no one has even been able to demonstrate an understanding of the error that Cantor made, much less address it.
We have but you have stubbornly refused to acknowledge it and correct the error of your own thinking.
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?

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.
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.


Again, do you deny that 0.25=0.250-0.2500?

What would be your decimal representation for 1/3?
" . . . 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

mgb
Guru
Posts: 1666
Joined: Sun Oct 03, 2010 1:21 pm
Location: Europe
Has thanked: 7 times
Been thanked: 21 times

Post #120

Post by mgb »

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...)

Post Reply