Why Cantor's Diagonalization Proof is Flawed.

[Divine Insight]

Here is a Youtube video, under 10 minutes, of Cantor's diagonalization argument.

Ok, I've seen this proof countless times.

And like I say it's logically flawed because it requires the a completed list of numerals must be square, which they can' t be.

~~~~

First off you need to understand the numerals are NOT numbers. They are symbols that represent numbers. Numbers are actually ideas of quantity that represent how many individual things are in a collection.

So we aren't working with numbers here at all. We are working with numeral representations of numbers.

So look at the properties of our numeral representations of number:

Well, to begin with we have the numeral system based on ten.

This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

How many different numbers can we list using a column that is a single digit wide?

Well, we can only list ten different numbers.

0
1
2
3
4
5
6
7
8
9

Notice that this is a completed list of all possible numbers. Notice also that this list is not square. This list is extremely rectangular. It is far taller than it is wide.

Let, apply Cantor's diagonal method to our complete list of numbers that are represented by only one numeral wide.

Let cross off the first number on our list which is zero and replace it with any arbitrary number from 1-9 (i.e. any number that is not zero)


[strike]0[/strike]
1
2
3
4
5
6
7
8
9

Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it.

Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using a diagonal line because the list is far taller than it is wide.

Now you might say, "But who cares? We're going to take this out to infinity!"

But that doesn't help at all.

Why not?

Well what happens when we make the next step? We need to make the list 2 digits wide now.

What happens?

Here is a 2-digit list of all possible numbers represented by 2 numerals.

00
01
02
03
04
05
06
07
08
09
11
12
13
14
15
.
.
.

95
96
97
98
99

What happened? Well, our completed list of possible numerals that is two digits wide has incrusted in vertical height exponentially. This list is now 100 rows tall and only 2 column wide.

Now let's cross off the first two digits of our list and replace them with arbitrary numerals.

[strike]0[/strike]0
0[strike]1[/strike]

Ok, for the first zero being stuck off the list, I'll chose to arbitrarily replace that with a 5. For the second digit being struck off the list I'll replace that arbitrarily with a 7.

My new number is 57.

Is 57 already on my completed list? Yes. It's just further down the list where I couldn't possibly reach it by drawing a diagonal line.

Now you might say, "But who cares? We're going to take this out to infinity!"

But duh? We can already see that in a finite situation we are far behind where we need to be, and with every digit we cross off we get exponentially further behind the list.

Taking this process out to infinity would be a total disaster.

You could never claim to have "completed" this process because you can't move down the list fast enough using a diagonal line that crosses off each digit diagonally.

The very nature of our system of numerical representation forbids this. You can't complete this process in a finite situation, and it gets exponentially worse with every digit you add to the width, then you could never claim to have completed this process by claiming to have taken it out to infinity.

"Completed Lists" of numerical representations of numbers are NOT SQUARE.

Yet Cantor claims to be creating a "Completed List" here. It's a bogus proof that fails. Cantor didn't stop to realize that our numerical representations of number do not loan themselves to nice neat square competed lists. And that was the flaw in his logic.

By the way you can't even do this using binary representations of numbers.

In Binary Representation

A completed list of binary numbers 2 digits wide:

00
01
10
11

It's not square. It's twice as tall as it is wide.

Add another digit it gets worse:

000
001
010
011
100
101
110
111

There is no way that a completed list of numbers can be represented numerically in square lists.

Yet Cantor's diagonal argument demands that the list must be square. And he demands that he has created a COMPLETED list.

That's impossible.

Cantor's denationalization proof is bogus.

It should be removed from all math text books and tossed out as being totally logically flawed.

It's a false proof.

Cantor was totally ignorant of how numerical representations of numbers work. He cannot assume that a completed numerical list can be square. Yet his diagonalization proof totally depends on this to be the case.

Otherwise, how can he claim to have a completed list?

If he's standing there holding a SQUARE list of numerals how can he claim that he has a completed list?

Yet at what point does his list ever deviate from being square?

It never deviates from being square. It can't because he's using a diagonal line to create it. That forces his list to always be square.

Georg Cantor was an idiot.

He didn't even understand how numerical representations of numbers work.

His so-called "proof" doesn't prove anything. It's totally bogus.

He can't claim to have a "completed list" by the way he is generating his list. Claiming to take this out to infinity doesn't help. With every new digit he creates he falls exponentially behind where he would need to be to create a "Completed List".

Yet that's what he claims to have: A Completed List.

It's a bogus proof, and I'm shocked that no mathematicians have yet recognize this extremely obvious truth.

They keep publishing this proof and teaching it like as is it has merit when in fact it's totally bogus.

[olavisjo]

∞
∑3 X 10^n-1 <-> ∑ 3 X 10^-n
n=1

Is this not a coherent concept?

[Divine Insight]

.

Take away your decimal point and you can do the same thing with integers.

1) 1
2) 12
3) 167
4) 5785


Ok, there's the beginning of my list of integers.

I need to make the integers on the right an extra digit long in every step just to facilitate the diagonalization method of elimination.

So now I run my diagonal line down that list


1) [strike]1[/strike]
2) 1[strike]2[/strike]
3) 16[strike]7[/strike]
4) 578[strike]5[/strike]

Replace the following

1 = 2
2 = 3
7 = 8
5 = 6

My new number is 2386

Therefore I've created an integer that's not on my list. And if I keep this up to infinity I will ALWAYS have a NEW integer that's not on my list.

It's numerical trickery.

It's not numerical trickery.

You have demonstrated that the set of natural numbers is uncountable, and for the same reason that the set of real numbers is uncountable.

Therefore it is wrong to conclude that the cardinality of the set of real numbers is greater than the cardinality of the natural numbers.

You have successfully demonstrated that Cantor's Diagonalization Proof is Flawed.

That's right. I have. Because we know that if we define "countability" as the process of being able to make a one-to-one and onto correspondence between two sets, then the natural numbers necessarily have to be countable by that definition. But since using Cantor's method we can show that we can even create integers that can't be accounted for, this proves that Cantor's method is false.

And I've even explained why it is a false method. It is a false method because completed lists of numerals cannot be square, and thus trying to create a complete list using a diagonal method can never work in any case. It can't even work with the integers. You'll always be able to create integers that can't be on your list.

However, there is something far deeper going on between the set of integers and the so-called set of real numbers. They are indeed intrinsically different qualitatively.

The integers (and even the rational numbers included) are all defined as the quantitative property of a set. Well, the rational numbers are defined as the ratio between two sets of cardinal quantities. But the point is that these definitions are all consistent. They are all based on a concept of cardinality.

However, the irrational numbers are not. They are not defined in terms of cardinality. Every irrational number that you can define in a meaningful way (i.e. not just proclaiming an endless string of random numerals to be an irrational number), is necessarily the result of a self-referenced situation.

Therefore irrational "numbers" are a totally different concept, than cardinal numbers. They are a totally different phenomenon.

Therefore, if you toss these "irrational numbers" in with the natural numbers and rationals to create a "new set" called the "Real Numbers", then you have indeed created a set that contains elements that have totally different qualitative properties from other members of the same set.

This would indeed result in a set that has a fundamentally different "quality" than the previous set of just natural numbers and rationals.

So it should not come as any surprise that these two sets are going to be fundamentally different in a qualitative sense.

My point is that we can understand precisely what's going on here by simply recognizing the true nature of "irrational relationships" and that attempting to treat them as "cardinal numbers" is naturally going to create absurdities.

If we truly want to be precise about what we are doing we should call the set of natural numbers and rational numbers the "Real Cardinal Numbers". And rename what we now call the "Real Numbers" to something like, "The set containing irrational relationships"

Trying to treat all real numbers as though they can be thought of as the cardinal property of a set only leads to absurdities such as larger and smaller cardinal infinities. Which is precisely the road that Cantor when down, and the whole mathematical community blindly followed him.

Well, like Keithprosser3 points out, not everyone has gone down that road entirely, there mathematicians who don't buy into these ideas of completed infinities.

However even those mathematicians have probably accepted the set of "Real Numbers" as being a valid idea of a collection of cardinal numbers. But that idea is wrong as well.

And it would actually be quite enlightening for the mathematical community to realize the true nature of irrational relationships and to stop thinking of them in terms of cardinality. That a wrong idea.

They are the result of self-referenced situations. They are not "absolute cardinal properties of any collections of individual things".

Like I had posted somewhere earlier. You can easily "push" the irrationality around in these situations thus proving that the irrational relationship belongs to the entire self-referenced situation and is not an absolute cardinal property of anything.

For example if you have a square with an irrational diagonal all you need to do is chose unity for the diagonal and suddenly it's the sides that have irrational lengths.

This irrational relationship doesn't belong to the diagonal, or the sides, it belongs to the whole object, the square. It's a self-referenced situation created by this situation in context. Same is true for Pi, and every other meaningful irrational number you can come up with. And this self-reference doesn't necessarily need to be geometric. There are many ways that situations can become self-referenced.

And that is the KEY in understanding irrational relationships.

It's also very enlightening because anytime you run into an irrational "number" you can be assured that there is indeed a self-referenced situation lurking somewhere in that relationship, and if you look for it you will find it.

So that's far more enlightening than just treating irrational relationships as though they are some kind of weird and incomprehensible cardinality of sets.

But current modern mathematicians have NO CLUE that irrational quantities always arise from self-referenced situations. They don't know this because they have chosen to treat irrational quantities as though they are cardinal properties of sets. They don't even realize this connection between irrational quantities and self-referenced situations.

[lao tzu]

So I would love to sit down with bunch of potentially interested mathematicians, and give a talk on this topic. Mathematicians who reject completed infinities would certainly be more promising that mathematicians who have accepted Cantor's baloney.

I have no desire to argue with mathematicians who are hell bent on defending Cantor's multiple sized infinities. If they are that determined to believe in that, then I'll just stay home and save myself the time and effort and they can believe in whatever they like.

I have no desire to argue with a bunch of Cantor Fans.

Dear DI,

Stay home, then. Even with concerted effort, your choice to substitute your own personal definitions in place of accepted terminology makes a discussion with you difficult. Picking out the errors that arise secondary to this decision is not so much a rhetorical exercise as a forensic undertaking. But many of us, especially those who teach, are used to dealing with these issues with our students. Still, we're not especially well suited for dealing with students making these mistakes, subsequently refusing to listen to explanation, and doggedly insisting on making the same mistakes again and again.

The personal insults constitute a non-starter, of course, but I'm assuming you could probably refrain in a direct conversation away from the internet.

What are these "completed infinities" you suppose mathematician's accept? It doesn't happen, of course, but I'd imagine you mean something by this phrase, and I'd like to know what it is. Ironically, I've seen completed infinities, sans caution quotes, aplenty in this discussion, all of which came from you, stemming from your linked contradictions: that an infinite array can be chopped off into a square, or more recently into a rectangle; and that an infinite string can be considered an integer. I've given you some reasons why any mathematician would reject them. There are many more. If you can find a mathematician that accepts either of these, I'd like to speak with them, because I don't presently believe they exist. I've never met one.

Hilbert's Hotel seems paradoxical to the uninitiated, but it is merely heuristic. It shows a property of infinite sets that is not shared by finite sets. You can't double the size of a finite set and still make the elements match up. But you can do this with any infinite set. We walked in knowing the finite and the infinite were different, and Hilbert's hotel shows us a specific instance of that difference. It's not the only difference; it's only an engaging difference that can serve to motivate further investigation.

Investigating further, we look at the other infinite sets we typically encounter. We know the real numbers and the rationals are different. To measure some of that difference, we've created the concept of cardinality.

Yes, it was born a century ago as Cantor's baby, but an entire host of mathematicians, including your correspondent, have been engaged in helping it grow up ever since. Dissing Cantor because you're uncomfortable with the theory of transfinite cardinals is akin to dissing Darwin because you're uncomfortable with evolution, or disputing the size of the pyramids because they were measured in a tutu. You can insult us freely, at least here in America where you're guarded by the first amendment, but it doesn't make the behavior any less rude or poorly considered as a critique of the arguments.

Keep in mind while you're sniggering at his voices from God that you're posting under the handle "Divine Insight."

The theory of cardinals is both a sharp-edged and a blunt tool. Blunt because it can't differentiate between the rationals and the counting numbers, but sharp in that it can find a difference between the reals and either one of them, or between the reals and any of the less well known sets we speak of as "countably infinite," such as the algebraic numbers I alluded to earlier.

We use these theories because they work. We replace these theories with theories that work better. Are you proposing something that works better? Well, no. Do you have, for instance, a measure or criterion that allows us to differentiate between both the reals and the rationals, and between the rationals and the integers? That's what you need.

It's not reasonable to expect anyone to trade in better for worse.

As ever, Jesse

p.s. Your method, a finite truncation of an infinite list you've named a "square" via some private definition — and it is your method, not Cantor's, or any mathematician's — is insufficient to count the counting numbers, even before we expand the set to include your 3forevers, 142857forevers, etc.

You could fix the first, by the way, by extending each of your terminating "integers" with an infinite string of zeros, though it'd fit like a tire patch on a bagel. You'd be better off simply sticking with Cantor's constructions, which wasn't broke until you broke it.

To accomplish the second, map the strings to associated simplified fractions, and order them first by the sum of numerator and denominator, and last by the numerator, yielding a total order on the full set of positive rational numbers.

1/1
1/2, 2/1
1/3, 2/2*, 3/1
1/4, 2/3, 3/2, 4/1
1/5, 2/4*, 3/3*, 4/2*, 5/1
1/6, 2/5, 3/4, 4/3, 5/2, 6/1
etc.

*skip unsimplified fractions to avoid redundancy

By observation and construction, then, the set of rationals is countably infinite. QED.

[keithprosser3]

Bravo!
For someone who has been dead for over 2000 years, you write a dam' good post, LT.

[Divine Insight]

The personal insults constitute a non-starter, of course, but I'm assuming you could probably refrain in a direct conversation away from the internet.

There are no personal insults coming from me. You are mistakenly taking my objections to an entire mathematical community personally. This would be like a Christian being personally offended when someone points out the absurdities in the Bible. Of course, Christians often are personally offended in this way, but it's totally unjustifiable since they didn't write the Bible.

In a very similar way, for you to become personally offended by objections to the development of the formalization of modern mathematics over last few centuries is truly absurd. You did not develop this formalism, and thus for you to become personally insulted over someone who rejects it is nothing other than absurdly silly.

For someone who calls himself "Lao Tzu" you should most certainly know better than to allow yourself to become emotionally insulted over someone's view of a formalism that you did not personally create.

What are these "completed infinities" you suppose mathematician's accept?

If you accept the work of Georg Cantor, you necessarily accept the concept of completed infinities because that's precisely what Georg Cantor's work is based upon. There were many famous mathematicians at the time who objected to this precisely on these grounds. Henri Poincare was one of the most vocal about this proclaiming Cantor's Set theory would be a disease from which the mathematical community would someday recover. I totally agree with Henri Poincare. Leopold Kronecker also rejected Cantor's completed infinities with a passion, as did many others.

So I'm just siding with a the mathematicians who reject this concept. And there were many of them at the time of Cantor, and there are still many today who do not accept these notions.

In fact, you appear to be one of them who believe that you can somehow accept Cantor's formalism whilst simultaneously rejecting his necessary requirement that infinities can be completed.

In fact, that's the only way in which infinities of larger or smaller cardinal properties can even exist.

If you accept infinity as an endless process, then what sense to you make of an infinite that is "larger" than this?

One set is more endless than another?

I hope you'll forgive any personal insult that this may seem to imply, but IMHO, that's absurd.

I totally reject the idea of completed infinities, and I totally reject the idea of infinities that are more endless than other infinities. That too is simply absurd.

Hilbert's Hotel seems paradoxical to the uninitiated, but it is merely heuristic. It shows a property of infinite sets that is not shared by finite sets. You can't double the size of a finite set and still make the elements match up. But you can do this with any infinite set. We walked in knowing the finite and the infinite were different, and Hilbert's hotel shows us a specific instance of that difference. It's not the only difference; it's only an engaging difference that can serve to motivate further investigation.

I have no problem with the concept of infinity as endlessness, or as an endless process. I'm not arguing against concept of infinity in general. A set that is endless is actually uncountable, IMHO. To claim that it can be used as a standard for countability (as Cantor claims) requires that it be thought of as a completed cardinality. And it is Cantor's proposal that I object to.

I have no problem with Hilbert's Hotel.

Investigating further, we look at the other infinite sets we typically encounter. We know the real numbers and the rationals are different. To measure some of that difference, we've created the concept of cardinality.

We didn't create the concept of cardinality to do this. On the contrary the concept of cardinality is older than the hills.

Moreover, there are two things wrong here:

One is that really really have no business creating a set called the "Real Numbers" and tossing irrational relationships into the set and attempting to treat irrational relationship as thought they themselves have well-defined cardinality. They don't.

So that was mistake #1.

However, given that mistake that seems historically inevitable to have happened, we still don't need the concept of cardinality to tell these two sets apart. Because the difference between them is no a difference in cardinality. They are both infinite sets and we truly only need one concept of infinity (i.e. endlessness)

Both of these sets are equally infinity. They are both endless sets. They do not have a difference in cardinal properties.

However, they are indeed quite different sets qualitatively speaking. The set of rational numbers is a set containing sets of valid cardinality.

The set of "Real Numbers" is a set that contains two different types of elements. One type of element are indeed the rationals which are themselves sets of valid cardinality, the other type of element are these irrational relationship that are not valid ideas of cardinality.

And therein lies the difference between these two infinite sets. The infinities are identically (i.e. they both contain an endless amount of elements). Neither set is more endless than the other. They are both endless.

The difference between these objects is a qualitative difference, not a difference in cardinality. And this is what becomes totally lost in Cantor's formalism.

Yes, it was born a century ago as Cantor's baby, but an entire host of mathematicians, including your correspondent, have been engaged in helping it grow up ever since. Dissing Cantor because you're uncomfortable with the theory of transfinite cardinals is akin to dissing Darwin because you're uncomfortable with evolution, or disputing the size of the pyramids because they were measured in a tutu. You can insult us freely, at least here in America where you're guarded by the first amendment, but it doesn't make the behavior any less rude or poorly considered as a critique of the arguments.

This has nothing at all to do with what I'm comfortable with.

I'm just making purely logical observations here.

Irrational relationship shouldn't be treated as cardinal numbers in the first place. It is neither necessary, not productive. On the contrary attempting to treat them as such leads to all sorts of utter absurdities that only serve to confuse the truth of what's going on.

You've compared this with Darwin's evolution, but that's a really bad comparison because there is much evidence for Darwin's evolution. Where is there any evidence that irrational relationship need to be thought of, or formally defined as cardinal properties of sets? That is a totally arbitrary choice what cannot be justified by nature.

In fact, this very notion brings up the question of whether or not you believe mathematics is a "science"?

So you believe that there is ONE TRUTH "out there" that mathematics reveals to us? And if so, then what criteria do you suggest we use to measure whether or not we actually have that ONE TRUTH?

And is that ONE TRUTH a physical property of this universe (like fossil records are for Darwin's theory of evolution)?

If you are going to compare the work of Georg Cantor with the work of Charles Darwin you had better be prepared to show physical evidence to back up Cantor's formalism.

I believe I can show just the opposite. I believe that I can show that Cantor's ideas are entirely whimsical (even he claimed that he got his ideas from God).

I can show physical evidence from this physical universe for my understanding of irrational relationships.

Therefore my theories are more akin to Darwin's than Cantor's are.

I'm suggesting that mathematics not only should be, but actually can be made into an actual physical science.

At least with respect to numbers and ideas of quantities. Please don't ask me to drag things like Boolean Algebra into this because Boolean Algebra has nothing at all to do with numbers as an idea of a cardinal property of a set.

I'm talking about mathematics here as a formalism of ideas of quantity which I propose is indeed an actual physical property of our physical universe.

Why do you think mathematics has worked so well when applied to the physical universe anyway?

Eugene Wigner, a physicist once wrote a paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Now what sense does that even make? Why should it be unreasonable that mathematics should be so effective in describing the natural sciences?

It would only be unreasonable if a person believed that mathematics was just a totally made-up whimsical idea of mankind (which it unfortunately has become recently).

But if mathematics were actually a science that stuck to only describing the actual quantitative physical behavior of the physical universe (as it first started out to be) then it should come as no surprise at all that mathematics describes the physical universe so well.

I claim that mathematics most certain can be a physical science. But in order to get there we need to back track and toss out places where we had made some horrible wrong turns.

If mathematics had stuck to it's guns from the very beginning it would be science. Unfortunately irrational relationship were discovered and the mathematicians didn't know how to handle them so they treated them like cardinal properties (which physically they are NOT).

And that was a wrong turn. That wrong turn was actually made thousands of years ago, and was already well-accepted by the time Cantor came along, so I guess we really can't pin the blame on Cantor for this underlying problem.

In any case, if you are going to hold that Cantor's work is equivalent to Darwin's work, then you had better be prepared to also claim that mathematics is a science that is directly dependent upon physical evidence from the physical world, and show me how that fits in with Cantor's work.

I am certain that you won't be able to do that, because that's not what we have here.

Cantor's ideas are entirely whimsical and he himself even claimed that he got his ideas from God, not from the physical universe.

Cantor's formalism is basically his own little religious whimsical cult.

Keep in mind while you're sniggering at his voices from God that you're posting under the handle "Divine Insight."

Let me tell you about my screen name.

When I came to logon I couldn't think of a decent screen name. I sat there at the computer and asked the Goddess to give me some divine insight into a clever screen name. I intuitively heard her say to me, "That's a good one".

So I typed it in and here I am.

But let's not get distracted into wishy-washy religious stuff.

The theory of cardinals is both a sharp-edged and a blunt tool. Blunt because it can't differentiate between the rationals and the counting numbers, but sharp in that it can find a difference between the reals and either one of them, or between the reals and any of the less well known sets we speak of as "countably infinite," such as the algebraic numbers I alluded to earlier.

But it can't. Cantor's diagonalization proof is logically flawed. His list of reals cannot be a complete list. Therefore the fact that he can come up with strings of numerals that can't be on his list is no surprise at all. It doesn't make any logical statements about the cardinality of any sets at all. It's a bogus proof.

But as I have already pointed out, mixing irrational relationships in with cardinal numbers in a single sit is already a bogus idea to begin with. So the whole shebang is flawed from square one.

We use these theories because they work. We replace these theories with theories that work better. Are you proposing something that works better? Well, no. Do you have, for instance, a measure or criterion that allows us to differentiate between both the reals and the rationals, and between the rationals and the integers? That's what you need.

But they don't work. They only seem to work in very esoteric situations concerning abstract mathematical collections that contain objects that don't even have well-defined cardinality themselves.

Moreover, there may be something useful in that kind of abstraction. I'm not saying that we can't studying those sorts of things. Just like I'm not saying that we shouldn't be studying Boolean Algebra. I'm simply saying that we need to recognize that these ideas are no longer ideas of proper cardinality, but instead they have become contaminated with other qualitative ideas.

Ideas that would actually be better studied if their own qualities were actually recognized to be something different from an idea of cardinality.

As I've already said, you can learn much more about irrational relationship when you realize their true physical nature than you can if you instead try to treat them like ideas of cardinality of sets.

It's not reasonable to expect anyone to trade in better for worse.

And I would never ask anyone to do that. What I have to offer is definitely better.

p.s. Your method, a finite truncation of an infinite list you've named a "square" via some private definition — and it is your method, not Cantor's, or any mathematician's — is insufficient to count the counting numbers, even before we expand the set to include your 3forevers, 142857forevers, etc.

You could fix the first, by the way, by extending each of your terminating "integers" with an infinite string of zeros, though it'd fit like a tire patch on a bagel. You'd be better off simply sticking with Cantor's constructions, which wasn't broke until you broke it.

To accomplish the second, map the strings to associated simplified fractions, and order them first by the sum of numerator and denominator, and last by the numerator, yielding a total order on the full set of positive rational numbers.

1/1
1/2, 2/1
1/3, 2/2*, 3/1
1/4, 2/3, 3/2, 4/1
1/5, 2/4*, 3/3*, 4/2*, 5/1
1/6, 2/5, 3/4, 4/3, 5/2, 6/1
etc.

*skip unsimplified fractions to avoid redundancy

By observation and construction, then, the set of rationals is countably infinite. QED.

I'm familiar with how the rationals can be placed into a one-to-one correspondence with the natural numbers. I'm not impressed, because there is not need to even do this anyway. It's utterly meaningless. Both the set of integers and the set of rationals are infinite.

But I can, and already did, I can show how using Cantor's method I can create infinitely many integers that cannot be put into a one-to-one correspondence with the integers themselves.

And there is not need to use infinitely long integers.

I already did it somewhere in this thread. Let me see if I can find it.

Here it is clear back from page #1 post #3:

Here is my list. On the left I have numbered the list with integers that I'm going to line up with. To the right of each number I choose an aribitary integer just like Cantor does in his diagonal argument.

I increase the numerical length of my integers in each row. But I am free to do this because these are arbitrary integers just as Cantor chooses his arbitrary reals. I could also always use longer integers if I so desire, but I do it this way simply because it makes for a neater diagram.

Here is my arbitrary list of integers

1) 1
2) 12
3) 167
4) 5785


So now I run my diagonal line down this list


1) [strike]1[/strike]
2) 1[strike]2[/strike]
3) 16[strike]7[/strike]
4) 578[strike]5[/strike]

I arbitrarily replace the following numerals.

1 = 2
2 = 3
7 = 8
5 = 6

My new integer is 2386, this integer cannot be anywhere on my list above.

In fact, I can create many more integers that cannot be on my list just by making different arbitrary substitutions above. And I can continue this forever.

What will I end up with? I will end up with infinitely many integers that cannot be anywhere on my infinite list.

Thus showing that I have created infinitely many integers that cannot be put into a one-to-one correspondence with the integers.

This is, of course, total baloney. But this is the folly of Cantor's diagonal method.

His method is clearly flawed. It can even be used to prove that the integers can't be put into a one-to-one correspondence with themselves which is totally absurd.

So this is a proof by contradiction that Cantor's diagonalization method of trying to put sets into a one-to-one correspondence with each other fails. It's logically flawed.

We can use his method to show that we can create a list of integers that can't be anywhere on our list whilst we are attempting to map them to the integers themselves.

And I've already explained why this is impossible. It's impossible because our system of numerical notation cannot be used to create square lists that are complete. This is why this method must always fail.

It has nothing to do with the set of real numbers being "cardinally larger" than the set of integers. They are both equally infinite.

[micatala]

I lost a previous reply to Divine Insight, so at this point I will summarize.


DI has not defined in any precise what he means by self-reference, or by a list of numbers being square.

ID has asserted that irrational numbers involve self-reference, and thus they are somehow problematical. In fact, irrational numbers can be defined in ways other than the diagonal of a square, etc. and so are not necessarily 'self-referential.'

DI says he accepts the idea of limits, as long as you don't assert the limit has to be somehow 'completed.' If that is the case, then Pi should be a perfectly unobjectionable quantity. Here are a number of ways to express Pi as a limit, most of these express Pi as the limit of rational numbers.

http://functions.wolfram.com/Constants/Pi/09/


DI says irrational numbers and even rational numbers do not designate cardinal quantities. My first response is "so what." This does not show they are somehow logically contradictory or absurd.

DI is laboring under the misapprehension that mathematicians think of irrational numbers as cardinal quantities, and that this somehow has something to do with the set of reals having a cardinality. This is like confusing the name of the nation with the names of the citizens within the nation.

DI says 33333333.... is an integer. This does not even meet his own definition. Pray tell, what set is having its cardinality designated by 333333.....?



DI is free to define his own private mathematics as he wishes. However, he is mistaken if he thinks his private definitions say anything about standard mathematics. It is as if he thinks that because someone can speak Spanish, that means anything one says in English is somehow absurd.


DI has yet to demonstrate any logical absurdities with Cantor's proof. He is laboring under a misapprehensions if he thinks his ill-defined notion of the 'squareness' of a list of numbers is essential to the proof. The essential notions of the proof are actually simple. One is that cardinality can be defined using bijective funtions, which DI at one point seemed to accept. The second is that expressions of the form (I'll do this informally)

Sum from 1 to infinity of a_i * 10^-i

correspond to real numbers, as this is really what we mean by the numeral

.a1 a2 a3 a4 . . . .

as a decimal expansion. This sum can be rigorously defined using DI's own terminology, ala Weierstrass, of trends and bounds. All we need for Cantor's proof along these lines is that infinite decimal expressions for numbers between 0 and 1 correspond to real numbers, and this that quite well.



As lao has pointed out, one could go through DI's posts point by point to point out the problems, but they do tend to fall into a few general categories. He does not precisely define his own terms. He asserts there are logical absurdities without any justification. He misunderstands numerous concepts of mathematics, thereby creating large straw men. He is inconsistent with himself in many cases.

At any rate, I will likely leave the thread for others to follow at this point. If DI wants to understand Cantor's proof, rather than making illogical criticisms of it, then I am willing to participate.

[Divine Insight]

I responded to everything you've addressed below micatala, however, if you'd like to just stick with the actual subject of this thread you can skip to the bottom and just address On the Details of Cantor's fallacious proof directly. After all, that is supposed to be the topic of this thread. Discussions about the validity of irrational cardinal numbers is actually an aside.

~~~~

DI has not defined in any precise what he means by self-reference, or by a list of numbers being square.

I disagree with your charge here. I have already addressed several instances of self-referenced relative relationships in very precise ways.

I have also address very precisely what I mean by a list of numbers being square. Cantor's diagonal list is necessarily always square. It has to be square because he is creating it by using a diagonal line that runs through the consecutive numerical digit of each numeral in his rows.

Therefore Cantor's list must necessarily always be square.

I have already shown why no complete numerical list of numerals can ever be square.

Therefore the fact that Cantor can produce a string of numerals that's not already on his square list above where he is currently working, is totally irrelevant. His square list can never be a complete list anyway. So his conclusion that he has created a number that can't be on a completed list is illogical. It's ungrounded in any logical sense.

All his method shows is that you cannot create a complete list of any numbers using his diagonal method to create the list.

What part of that do you not understand?

You say that I'm not being precise enough, but I have already shown everything required to back up my observations.

I can't be anymore precise than that.

DI has asserted that irrational numbers involve self-reference, and thus they are somehow problematical.

I never said they were "problematical", I simply said that they do not represent an idea of cardinality of a set. They do not satisfy the formal definition of a cardinal number as a quantitative property of a collection of individual things.

In fact, irrational numbers can be defined in ways other than the diagonal of a square, etc. and so are not necessarily 'self-referential.'

I hold that every well-defined irrational number can be shown to be the result of a self-referenced situation. I reject any claim that you can create a well-defined irrational number by writing out an endless string of numerals for infinity because you cannot complete that process and therefore you have not created a well-defined cardinal quantity.

DI says he accepts the idea of limits, as long as you don't assert the limit has to be somehow 'completed.' If that is the case, then Pi should be a perfectly unobjectionable quantity. Here are a number of ways to express Pi as a limit, most of these express Pi as the limit of rational numbers.

http://functions.wolfram.com/Constants/Pi/09/

Limits do not qualify as a meaningful way to define the existence of a cardinal number. Weierstrass made certain that this could not be done in his epsilon/delta definition of the Calculus Limit.

So showing me all these limits that approach Pi, is totally irrelevant.

Moreover, have you even looked at those limit expressions? I don't see a single solitary one that doesn't contain a self-referenced situation in the variable n.

DI says irrational numbers and even rational numbers do not designate cardinal quantities. My first response is "so what." This does not show they are somehow logically contradictory or absurd.

I never said it did.

I simply said that if they do not represent cardinal quantities then we shouldn't be treating them as though they are cardinal numbers.

DI is laboring under the misapprehension that mathematicians think of irrational numbers as cardinal quantities, and that this somehow has something to do with the set of reals having a cardinality. This is like confusing the name of the nation with the names of the citizens within the nation.

No, that's not my position at all. I'm not saying that mathematician think of irrational relationships as cardinal numbers because they have placed them into a set of numbers called "Real Numbers".

I'm saying just the opposite. They created such a set in the first place precisely because they were already treating irrational relationships as though they can be treated as cardinal numbers.

DI says 33333333.... is an integer. This does not even meet his own definition. Pray tell, what set is having its cardinality designated by 333333.....?

I confess, that this was a very bad mistake on my part. All natural numbers must necessarily be finite. But I have pointed out that this is itself a paradox if we demand that the set of natural numbers must be infinite. The only way that the set of natural numbers can be infinite is if we allow for integers to grow without bound. And the only way they can do that is if we demand that there can be no largest integer. And by making that single demand we have created a situation where integers themselves must necessarily become infinite.

So there's already a paradox in the definition of the set of Integers anyway.

DI is free to define his own private mathematics as he wishes. However, he is mistaken if he thinks his private definitions say anything about standard mathematics. It is as if he thinks that because someone can speak Spanish, that means anything one says in English is somehow absurd.

I don't think that my definitions say anything about standard mathematics.

I'm saying that the definitions used by standard mathematics are "wrong" relative to the true quantitative properties of our physical world.

This also doesn't mean that I don't understand the standard definitions of mathematics. I do understand them, and this is why I can clearly see that they are wrong. Not only can I show that they are wrong but I can even point to precisely where they have made a mistake when they created the definition historically.

DI has yet to demonstrate any logical absurdities with Cantor's proof.

I have already proven that Cantor's proof is invalid.

Cantor's diagonal list must be a square list because of the very means by which he is creating it. Therefore it cannot be a complete list of numbers in numerical notation. The fact that he can come up with numbers that can't be on his square list is meaningless, and cannot be used to prove the conclusion that he claims it proves.

He is laboring under a misapprehensions if he thinks his ill-defined notion of the 'squareness' of a list of numbers is essential to the proof. The essential notions of the proof are actually simple. One is that cardinality can be defined using bijective funtions, which DI at one point seemed to accept.

Only for finite sets. It makes no sense to use this to try to define the precise cardinality of an infinite set.

The second is that expressions of the form (I'll do this informally)

Sum from 1 to infinity of a_i * 10^-i

correspond to real numbers, as this is really what we mean by the numeral

.a1 a2 a3 a4 . . . .

as a decimal expansion. This sum can be rigorously defined using DI's own terminology, ala Weierstrass, of trends and bounds. All we need for Cantor's proof along these lines is that infinite decimal expressions for numbers between 0 and 1 correspond to real numbers, and this that quite well.

And I've already shown that Weierstrass made certain via his formal definition of the calculus limit that you cannot use a limit to proclaim the existence of the quantity defined by the limit.

All you can say is that if you could complete this process you could construct this number. But a calculus limit does not require, nor suggest that you can complete this process.

This is actually calculus being used in poor form. And this is even true accepting standard mathematics just the way it exist. I don't need to add anything new here.

As lao has pointed out, one could go through DI's posts point by point to point out the problems, but they do tend to fall into a few general categories. He does not precisely define his own terms.

Not true. I do define my own terms.

He asserts there are logical absurdities without any justification.

Not true. I gave sufficient justifications for all of the logical absurdities that I have pointed to.

He misunderstands numerous concepts of mathematics, thereby creating large straw men. He is inconsistent with himself in many cases.

You'll need to be more specific than this. You have already jumped to several wrong conclusions about my position on various things. So you need to be sure that you fully understand what I'm actually saying before you jump to conclusions that I'm wrong about anything.


On the Details of Cantor's fallacious proof

At any rate, I will likely leave the thread for others to follow at this point. If DI wants to understand Cantor's proof, rather than making illogical criticisms of it, then I am willing to participate.

I am absolutely positive that Cantor's proof is logically flawed. I've actually proven this fact in this thread, but you have refused to follow that proof.

What part of my proof do you not understand?

Here's a summary of my proof.

Observation #1. Completed lists of numbers represented by numerical representations cannot be square lists.

Do you understand why this is so? Or do you need for me to go over this again?

Observation #2. Cantor's numerical list is necessarily square because he is creating it using a diagonal line that crosses over every new digit in every new row. His list necessarily must be square at all times. It can never deviate from being square.

Observation #3. Cantor then comes up with a new number that cannot be above the point where is he working on his square list, and then proclaims that this number therefore cannot be anywhere on a completed list.

But #3 is not true. The number he had just created simply cannot be on his square list above where he is currently working, but it does not follow that it cannot then be on an actual completed list which cannot be square.

Cantor's diagonalization proof is grossly flawed.

All he has done is demonstrate that a square list or numerals cannot be a complete list of numbers. But we already know this.

So he hasn't proven anything.

Also, keep in mind that even if it could be shown using some other method that the real numbers cannot be put into a one-to-one correspondence with the Integers this still doesn't fix Cantor's proof.

In other words, even if he accidentally arrived at a true conclusion, his method of proof is still logically flawed and invalid. It is a bogus proof no matter what. Period.

And it's flawed within the standard rules of mathematics. You don't need to accept any new definitions or anything. You can totally reject all of my objections concerning irrational relationships and all the rest. Cantor's diagonalization proof is still grossly flawed.

So which of my 3 observations above do you disagree with?

[JohnA]

I agree here with DI. Cantor was a loony, and his rubbish is a disease. I hope mathematicians would wake up and purge it.
The mere fact that Cantor reserved absolute infinity for his god, but also stated that this absolute infinity has mathematical properties, says it all.

[micatala]

I responded to everything you've addressed below micatala, however, if you'd like to just stick with the actual subject of this thread you can skip to the bottom and just address On the Details of Cantor's fallacious proof directly. After all, that is supposed to be the topic of this thread. Discussions about the validity of irrational cardinal numbers is actually an aside.

Fair enough. I will skip to the bottom.

On the Details of Cantor's fallacious proof

At any rate, I will likely leave the thread for others to follow at this point. If DI wants to understand Cantor's proof, rather than making illogical criticisms of it, then I am willing to participate.

I am absolutely positive that Cantor's proof is logically flawed. I've actually proven this fact in this thread, but you have refused to follow that proof.

What part of my proof do you not understand?

Here's a summary of my proof.

Observation #1. Completed lists of numbers represented by numerical representations cannot be square lists.

Do you understand why this is so? Or do you need for me to go over this again?

Your argument here is incorrect for a whole host of reasons.


First, you have only vaguely defined what it means for a list of numbers to be square.

Second, and more importantly, you have asserted repeatedly without any proof that completed lists of numbers cannot be square.

Here is an example of such an assertion from page 4 or so of the thread.

Cantor's list is square. Period.

I've already shown, using totally valid mathematical truths, that a square numerical list of numbers can never be complete. It's extremely impossible using the decimal or tens system. And it's even impossible if you go all they down to binary, the simplest possible numerical system we have.

No complete list of numerals can ever be square.

That's a mathematical truth that holds no matter what.

This is actually a mild improvement over earlier assertions. However, despite your many repeated assertions, you have never shown that whatever you mean by a completed list of numbers cannot be square.

Finally, and this is really the clincher, you are mistaken in your assertion that having a completed square list of numbers is really the crux of Cantor's proof. As has been explained, Canto's proof proceeds by contradiction, makes use of the correspondence between real numbers and infinite decimal expansions, and makes use of the definition of cardinality based on one-to-one correspondences.



Now, if you wish to elaborate (please do not simply repeat your previously unsubstantiated assertions) what you mean by square lists and completed lists fine, but again, none of that is relevant unless you can explain how your notions actually address Cantor's proof. Nowhere in his proof does he use those notions. If you think they are relevant, you need to show how using the notions he actually does use.

I'm familiar with how the rationals can be placed into a one-to-one correspondence with the natural numbers. I'm not impressed, because there is not need to even do this anyway. It's utterly meaningless. Both the set of integers and the set of rationals are infinite.

Here, you accept the rationals and integers can be placed in one-to-one correspondence. The fact that you are not impressed has no logical validity. Neither do your assertions about what is meaningless. One main problem with your argument is the high degree of subjectivity you bring to it.

But I can, and already did, I can show how using Cantor's method I can create infinitely many integers that cannot be put into a one-to-one correspondence with the integers themselves.

As I recall, you did his using your infinite integer '333333.....' which you now admit was a major mistake.



Observation #2. Cantor's numerical list is necessarily square because he is creating it using a diagonal line that crosses over every new digit in every new row. His list necessarily must be square at all times. It can never deviate from being square.

Cantor does make use of the diagonal. However, the key issue is not 'squareness' but the correspondence between real numbers and infinite decimal expansions. As I noted earlier, those decimal expansions are entirely consistent even with your notion of calculus and limits.

Do you have a refutation of the fact that every infinite decimal expansion of the form 0. a1 a2 a3 a4 . . .

corresponds to a real number and that every real number has at least one such decimal expansion (I say at least one, since, for example 1.0000 . . = 0.999999. . . .)

Observation #3. Cantor then comes up with a new number that cannot be above the point where is he working on his square list, and then proclaims that this number therefore cannot be anywhere on a completed list.

But #3 is not true. The number he had just created simply cannot be on his square list above where he is currently working, but it does not follow that it cannot then be on an actual completed list which cannot be square.

Cantor shows there is a number that is not on the list. And remember, he is not proving such a list exists, he is proceeding by contradiction. So, assumes for the purposes of contradiction that all real numbers between 0 and 1 can be put into one-to-one correspondence with the natural numbers, and then shows this cannot be the case. Your 'above the point where he is' is not relevant. It is basically, like your assertions in item 1, a straw man.

In the second paragraph, are you asserting there is a completed list which is not square? This is not making any sense, and together with your 'above where he is' invalidates your refutation since you are not actually even addressing, it seems, Cantor's actual argument.

Cantor's diagonalization proof is grossly flawed.

You have yet to show how in any coherent way.

All he has done is demonstrate that a square list or numerals cannot be a complete list of numbers. But we already know this.

If, by complete list of numbers, you mean a set in one-to-one correspondence with the rationals, we do know this, and we know it because of Cantor's proof.

Also, keep in mind that even if it could be shown using some other method that the real numbers cannot be put into a one-to-one correspondence with the Integers this still doesn't fix Cantor's proof.

That is true, but not particularly relevant.

In other words, even if he accidentally arrived at a true conclusion, his method of proof is still logically flawed and invalid. It is a bogus proof no matter what. Period.

Again, you have yet to give any logically coherent argument to show this.




Which part of Cantor's actual proof to have objections to?

Do you not accept that if there is a bijective function f:A to B that A and B are of the same cardinality?

Do you not accept that infinite decimal expansions as described above correspond to real numbers between 0 and 1?

[Divine Insight]

First, you have only vaguely defined what it means for a list of numbers to be square.

Ok, maybe square isn't even the proper terminology. We could even talk about triangular lists.

My point is that any completed list of number represented in numerical notation must be extremely rectangular. They can neither be square, nor triangular. They must be extremely rectangular being far taller than they are wide.

We're talking about "completed" lists here.

Second, and more importantly, you have asserted repeatedly without any proof that completed lists of numbers cannot be square.

Ok, let's do this one thing, and just stick with it for now until we come to some sort of conclusion concerning this one issue.

Here is my argument that any "completed" list of numerals, must be far taller than it is wide.

Let's start with the natural numbers.

Remember we are using numeral notation here and we are seeking a "completed list" of all possible numerical notations of the natural numbers.

Let's begin by listing the counting numbers from 1 to 9.

1
2
3
4
5
6
7
8
9

This is a "Completed List" of single digit numbers that can be represented using numerical notation. (you could add zero at the top of the list if you care to).

There is no single digit numeral that represents a number higher than 9. So this is a "Completed List" using this numerical notation.

Now how are you going to go down this list with a diagonal line to create a new number that isn't already on this completed list?

If you start by crossing out the first digit like so:

[strike]1[/strike]
2
3
4
5
6
7
8
9

And replacing it with an arbitrary numeral other than itself (2 thru 9)

You will have created a new number that is "Not yet on your list". However, that number would clearly already be on the "completed List" above. You just couldn't go down the list fast enough using a diagonal line that crosses over to a new digit with every new row that you create.

Let's move up a digit. Take a list that 2 digits wide. How tall would a "completed List" need to be.

Well, let's take a look. Here's the list. I'll just start with 10 because we already saw the problem with the single digit "Completed List" list above this.

10
11
12
13
14
15
16
17
18
19
20
.
.
.
95
96
97
98
99

It's going to be roughly 2 digits wide and about 90 rows tall (not counting the previous 9 rows above this)

Can you create a number using a diagonal line that is not on this "Completed List"?

No you cannot.

Let's try.

Here we go (I'll only list what I need to in order to create my new number using a diagonal line)

[strike]1[/strike]0
1[strike]1[/strike]

Ok, now replace both of the struck off digits with any single-digit numeral you like that is different from the numeral you struck off.

Is the new number you came up with already on the completed list above? Yes, it absolute is because that list is indeed "complete". In contains every possible number that can be represented using numerical notation.

So what do we do now? Well if we want to continue using a diagonal line to construct our supposedly "NEW" number that isn't on a completed list we have no choice but to expand to 3 numerical places. Now we'll be creating a supposedly "NEW" number that isn't already on our competed list. A list this wide is already 1000 rows deep at this point, and it's only 3 digits wide!

Completed lists of numbers in numerical notation necessarily must be far taller than they are wide, and this fact increases exponentially with every digit added to the width.

Therefore to create a new number using a diagonalization method and numerical notation at the same time is futile. It cannot be used to prove anything at all. Except perhaps that you can appear to be creating numbers that cannot be on the list above where you are currently working. But what would that prove?

Let me continue on with a few more points that are directly related to the above:

Cantor shows there is a number that is not on the list. And remember, he is not proving such a list exists, he is proceeding by contradiction. So, assumes for the purposes of contradiction that all real numbers between 0 and 1 can be put into one-to-one correspondence with the natural numbers, and then shows this cannot be the case. Your 'above the point where he is' is not relevant. It is basically, like your assertions in item 1, a straw man.

But Cantor's proof is meaningless because he could never go down any "REAL LISTS" fast enough using his diagonal line method.

Besides, I can use his same method to prove that the Integers cannot be put into a one-to-one correspondence with themselves, and I do that next just for completeness:

But I can, and already did, I can show how using Cantor's method I can create infinitely many integers that cannot be put into a one-to-one correspondence with the integers themselves.

As I recall, you did his using your infinite integer '333333.....' which you now admit was a major mistake.

I don't need the dots at the end. Here we go, I'm going to do precisely what Cantor is doing:

Here are my arbitrary choices to correspond the natural numbers to themselves.

1: 9
2: 23
4: 345
5: 2896
6: 87639


I'm just lining up arbitrary whole numbers with every natural number, just like Cantor lines up arbitrary real decimal expansions with the natural numbers.

Sure, I'm using numbers that increase in digits with every row. But I don't need to do that. Could have just as easily chosen the following list.

1: 923490568363840039
2: 45
3: 290349085688309487
4: 100000000000000000
5: 93847569930
6. 034948

These are just arbitrary integers no different from Cantor's arbitrary decimal reals.

So now let's go down this list with a diagonal line and see what happens.

1: [strike]9[/strike]23490568363840039
2: 4[strike]5[/strike]
3: 29[strike]0[/strike]349085688309487
4: 100[strike]0[/strike]00000000000000
5: 9384[strike]7[/strike]569930
6. 03494[strike]8[/strike]

Well what can we do? to begin with we can just go right down this line and replace each numeral with the following 123456.

Now we have a new number that is not on our list.

We can also create a new number 234567 and so on. We can actually create a whole list of new integers that cannot be on our list as we go.

And we can continue this for as long as we wish. In fact, in theory there is absolutely no reason why we should ever have to stop, and heaven forbid if we did have to stop, because if we actually had to stop we'd be standing there with a whole big long list of natural numbers that cannot be on our list!

Why does this work? Have we just proven that the natural numbers cannot be put into a one-to-one correspondence with themselves? Of course not.

The reason this works is because it's a faulty proof. It doesn't prove anything other than completed lists of numerals cannot be "Square" or "Triangular". They must necessarily be very tall rectangles.

But the very means of creating new numbers that supposedly can't be on this list by using a diagonal line forces the list to be square (or at least triangular) at every stage of the process. And this is why the proof is absurd.

Now you might argue that Cantor is using decimals that are padded to the right with zeroes. But that doesn't change a thing.

This method is simply wrong. It's not a sound logical argument.

Cantor failed to take into account the nature of numerical representations of number. Square lists (or even triangular lists) cannot be complete lists. It's that simple. A completed list must necessarily be extremely tall compared to the width of it's members. It can neither be square nor triangular. It is necessarily a very tall rectangular list containing many more rows than it is digits wide.

Therefore Cantor's list cannot be a complete list no matter how far he takes this process out. In fact, he gets exponentially further behind any possible completed list with every new digit he scratches off. So even if he could complete his infinite process he would be infinitely behind as well.

Taking a process where you are exponentially "losing ground" to infinity isn't going to help matters.

Cantor's diagonal process is exponentially "losing ground". He's exponentially falling further behind with every new digit he scratches off with his diagonal line.

The slope of his diagonal line simply isn't steep enough, and it could never be made steep enough because of the very nature of numerical representations.

Completed lists of numbers in numerical notation are exponentially taller than they are wide. No diagonal method of constructing a new number that's "not on the list" could ever hope to succeed.