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.

[lao tzu]

0.23234792340340584359834502408234098734953734535930087..... and so on.

The digits are entirely random, produced by rattling my numeric key pad.

I haven't thought it through myself, but I don't see any self-referencing going on at first glance. If DI demolishes that irrational number, there are plenty more. There are after all more irrational numbers than there are rational numbers, so we could be at this for some time!

Dear Keith,

You're working too hard. Any decimal number that does not terminate in zeros will do.

As ever, Jesse

[Divine Insight]

0.23234792340340584359834502408234098734953734535930087..... and so on.

The digits are entirely random, produced by rattling my numeric key pad.

I haven't thought it through myself, but I don't see any self-referencing going on at first glance. If DI demolishes that irrational number, there are plenty more. There are after all more irrational numbers than there are rational numbers, so we could be at this for some time!

I don't need to demolish it. It doesn't fit the criteria:

I said, "I would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation."

What is the meaning of your arbitrary string of numerals? What is the meaningful quantity that it supposedly represents?

Numeral are NOT numbers. They are just symbols that represent certain quantities.

For you to just arbitrarily type out a bunch of numerals and proclaim that this is a meaningful quantity is nonsense.

That would be no different from me typing out an arbitrary string of letters like "alghlkjdsgdslkh" and claiming that this then must represent a legitimate and meaningful word just because it uses the symbols that are used to convey ideas via words.

So I reject your arbitrary string of digits as having no "meaningful quantitative" value.

Moreover, since you are typing this out you must do this FOREVER. For if you were to stop at any moment you would suddenly have a RATIONAL NUMBER.



In fact, you must type FOREVER in order to produce your supposedly irrational number, because every number you type is a rational number at that point.

When do you ever cross the magic line to the point where you have actually typed an irrational number? NEVER! You never cross that line. With every digit you type you currently, at that point, necessarily have a rational number. So when you believe you can cross over that point to actually claim to have your irrational number?

Good luck with trying to produce a legitimate irrational number by constructing it one numeral at a time. AT EVERY STEP in your process, you have a rational number.

[Divine Insight]

I would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation.

Dear DI,

The diagonal length of a unit square is an irrational quantity.

As ever, Jesse

And that's based on a self-referenced geometric situation just as I had said.

No, it's not.

You'll need to do better than that.

You need to actually produce this quantity without using a self-referenced situation.

Thus far you you haven't shown how that's possible.

The length of the diagonal of a square is only irrational relative to the square itself. That's a self-referenced situation.

[Divine Insight]

Please tell me which real number I've missed.

There are an uncountably infinite number of real numbers missing from your list, but we don't need to go beyond the rationals to find one.

You missed 1/3.

No, I didn't miss it.

Here it is right here.

33333333333... : 0.33333333333...

[lao tzu]

Please tell me which real number I've missed.

There are an uncountably infinite number of real numbers missing from your list, but we don't need to go beyond the rationals to find one.

You missed 1/3.

No, I didn't miss it.

Here it is right here.

33333333333... : 0.33333333333...

Dear DI,

Whatever that string of characters may be, it is most certainly not a counting number. Counting numbers always, without exception, have successors and predecessors, for instance.

That thing doesn't.

Regards, J

[lao tzu]

You'll need to do better than that.

Dear DI,

On the contrary, you do.

You need to actually produce this quantity without using a self-referenced situation.

I just did.

Thus far you you haven't shown how that's possible.

Construct a line segment from two opposite vertices of a square. This segment is called the diagonal.

The length of the diagonal of a square is only irrational relative to the square itself. That's a self-referenced situation.

The diagonal, itself, is not the square, and hence does not refer back to itself. Are you familiar with the term self-referential? I have reason to doubt this now.

A square consists of four congruent line segments which meet at right angles to form a closed curve. It is entirely independent of its diagonal, its center, its area, its shadow, how fast we spin it, the color we may paint the interior, or any other quantity other than the four congruent line segments which meet at right angles to form a closed curve.

You are familiar with squares, aren't you? This is the second time on this thread you've given me reason to doubt.

Regards, J

[Divine Insight]

Please tell me which real number I've missed.

There are an uncountably infinite number of real numbers missing from your list, but we don't need to go beyond the rationals to find one.

You missed 1/3.

No, I didn't miss it.

Here it is right here.

33333333333... : 0.33333333333...

Dear DI,

Whatever that string of characters may be, it is most certainly not a counting number. Counting numbers always, without exception, have successors and predecessors, for instance.

That thing doesn't.

Regards, J

Sure it does.

Surely you can count up to 3.
How about 33?
How about 333?
How about 3333?
How about 33333?
How about 333333?
How about 3333333?

Ect.

What is ever going to cause you to stop?

So clearly 33333333333333333333333333... (continued as far as you like), has both a predecessor and a successor.

There is no reason that this process must ever terminate.

This is why the set of Integers is infinite (i.e. endless).

This process clearly has a predecessor as you can always count up to where your dots begin.

And the process clearly always as a successor because you can always continue counting beyond any point where you have temporarily stopped.

This is precisely why the set of Integers is infinite to being with.

We already have an infinite set. There is no reason to pretend that we need a larger infinity.

Infinity = endless. Period.

There is no such thing as larger and smaller infinities. That's Cantor's nonsense.

[micatala]

I appreciate Divine Insight at least making an attempt to understand mathematics in general and Cantor in particular. I may attempt further debunking of his arguments, such as they are, and I appreciate lao Tzu's contributions here as well.



However, not to be unkind, but the types of arguments being given here by DI are typical of what might be found in the following interesting and amusing book.


Mathematical Cranks, by Underwood Dudley.


The book certainly covers much more extreme versions than what we have seen here, but the bottom line is, DI is essentially creating his own ad hoc definitions of common mathematical concepts, is not using definitions in precise ways, and is making assertions without any logical support for them.


For example, lao Tzu is correct, DI's matching function does miss 1/3. Integers with infinitely many digits might exist as strings but not as numbers.

I am surprised DI would even seriously suggest this given his focus on defining numbers via cardinalities.



In addition, I find it astonishing that DI would say this:

It also doesn't help Cantor's proof that the real numbers are NOT countable.

and then a few lines later in the same post this:

But they are countable. Especially the list of Reals that Cantor was working with from 0 to 1.

http://debatingchristianity.com/forum/v ... 619#607619


Is DI now asserting that the real numbers, or at least the interval (0, 1), both is and is not countable?????

[Divine Insight]

but the bottom line is, DI is essentially creating his own ad hoc definitions of common mathematical concepts.

Yes, I am creating my own definitions. I wouldn't call them "ad hoc" though since they have very solid foundations in logic and reason.

And I am rejecting the formal definitions that have already been accepted by the current formal mathematical community. That's absolutely true. In fact, if I had to accept their definitions verbatim then I'd have no choice but to also accept their conclusions.

It's their definitions that I object to.

is not using definitions in precise ways,

But I am using definitions in precise ways. I'm just not accepting the modern formal mathematical definitions. (Note: I confess that there is a caveat to this which I address below).

After all, surely even you must concede that I am up against an extremely hostile crowd. I am up against people who are doing to defend the status quo to their deaths. They are going to defend the current modern formal definitions. And because of this they are convinced that I necessarily have to be wrong.

And clearly I would be "wrong" if I too had to accept all of the definitions as laid out by modern mathematics, because after all, modern mathematics has already demanded that irrational relationship must be accepted and treated as cardinal properties of sets (i.e. treated as cardinal numbers). Which I strongly object to.

In fact, that's my whole objection right there.

and is making assertions without any logical support for them.

I have logical support for all my assertions. But I cannot support this using modern day mathematical definitions because my claim is that those definition are already logically flawed.

For example, lao Tzu is correct, DI's matching function does miss 1/3. Integers with infinitely many digits might exist as strings but not as numbers.

I am surprised DI would even seriously suggest this given his focus on defining numbers via cardinalities.

Well, don't be too surprised by this, because that's where the caveat lies.

If you recall I reject the whole idea of treating "real numbers" (or in particular irrational numbers) as though they can be thought of as cardinal numbers to begin with. So, in truth, the very idea that this absurd "set" of objects needs to even be considered at all, much less placed in a one-to-one correspondence with the imagined collection of all integers is itself a totally unnecessary and absurd endeavor.

Moreover, how would you even hope to even define the "complete set" of irrational numbers? Such a set is ill-defined to begin with.

In addition, I find it astonishing that DI would say this:

It also doesn't help Cantor's proof that the real numbers are NOT countable.

and then a few lines later in the same post this:

But they are countable. Especially the list of Reals that Cantor was working with from 0 to 1.

http://debatingchristianity.com/forum/v ... 619#607619

Is DI now asserting that the real numbers, or at least the interval (0, 1), both is and is not countable?????

And you're right, I shouldn't have even gotten into that whole mess, because if you recall, I reject the very notion of even treating these so-called "real numbers" as though they can be thought of a cardinal properties of sets.

So I shouldn't even be engaging in any imaginary "proofs" of how they might be placed in a one-to-one correspondence with the integers.

~~~~~

The "set" of integers already has it's own paradoxes.

We say that the set of integers is "infinite". This is because we can always create a new integer from a preexisting integer by simply adding 1 to it.

Yet at the same time we imagine that every single integer is necessarily a quantitative property of finite cardinal set.

This is already a paradox.

In other words, we are claiming that the set of integers (which itself is a set of sets where each set within this set is an integer representing a finite cardinal quantity of its own individual set), represents an overall infinite set.

In other words, we don't see any reason that would restrict us from continually adding more and more integers to this set of integers endlessly. Thus we "intuitively conclude" that this set must then itself have the property of containing an infinite number of elements.

But is that genuinely rational logic?

If we are simultaneously demanding that every set within the set of integers (i.e. every integer itself) must necessarily be finite, then how can we truly claim that the "completed" set of integers would be "infinite".

Well, in truth we can't. That's not a logical conclusion itself. That's an absurdity.

We can only handle this in one of two ways.

We can either say that the set of integers itself must be finite since every integer within it must be finite, OR, we can handle it intuitively be simply saying that this set of integers has the potential for endless construction.

In other words, the set of natural numbers has the potential for endless construction, but it's actually wrong to say that it "contains" an infinite number of elements, and then imagine this as a "completed set".

Yet this is precisely what Georg Cantor (and now the whole rest of the mathematical community is trying to do).

So yes, if we are going to demand that every single element within the set of integers is itself a finite set, then I'm not convinced that we even have the logical "right" to demand that the set itself must then be infinite.

The only way it could be infinite is if we allow the elements within it to become large "without bound". In other words, we must allow there cannot be a "largest" integer". But the only way that could be true is if they integers themselves are without restriction on being finite.

So there is already a paradox just with the set of integers.

For us to think we are moving "forward" by then creating new "cardinal numbers" that are the result of self-referenced situations, where each of these sets cannot even be finitely defined, is just adding paradoxes on top of paradoxes.

Moreover, it's not even necessary. And that's my whole point.

We simply don't "need" to even go there.

It just not necessary. It's confusion that isn't required. We can live without just fine and actually understand irrational relationship MUCH BETTER by recognizing them for what they truly are instead of pretending that they can be treated like as if they are cardinal properties of sets (which they most certainly are not).

~~~~~~

And Cantor's diagonalization proof is logically wrong no matter what.

A square list of numerals (even decimal numerals) cannot be complete. Yet Cantor's proof requires that his list is complete. Otherwise, the number he has generated COULD still be on a complete list. The fact that it's NOT anywhere on his limited square list is totally irrelevant and utterly meaningless.

So his so-called "proof" is wrong no matter what. Even if you accept irrational numbers as cardinal numbers Cantor diagonalization "proof" is still bogus and logically flawed in any case.

Even if it's true that the "reals" (as defined by using endless decimal notion) cannot be placed into a one-to-one correspondence with the integers, Cantor's diagonalization proof is still locally flawed and does not prove this result. Even if he did accidentally come to the correct conclusion that doesn't make his proof valid.

A square numerical list cannot be complete, as I have already shown. But Cantor's diagonalization proof requires that his list be complete. So it's a failed proof. His list cannot be a complete list. Therefore if he used his list to generate a new number that's not on "his list" that's totally irrelevant and doesn't prove anything.

So it's a bogus proof no matter what.

[lao tzu]

Sure it does.

Surely you can count up to 3.
How about 33?
How about 333?
How about 3333?
How about 33333?
How about 333333?
How about 3333333?

Ect.

What is ever going to cause you to stop?

So clearly 33333333333333333333333333... (continued as far as you like), has both a predecessor and a successor.

There is no reason that this process must ever terminate.

Dear DI,

Watch carefully:

1
2
3
33
333
3333
33333
...

Please note your newly revised "counting" algorithm contradicts your previous counting algorithm, and now presents itself as a set of counting numbers that never arrives at 4.

The entire argument was absurd, as it's been known for a century now that the integers cannot be made to line up 1-1 with the real numbers. Redefining the integers by inserting enough other objects to bring them up to size ... well, that's cheating. It didn't work, either, even for the rational numbers. You've left yourself no way to handle rationals that repeat after 2 digits, or 3 digits, and so on. Not to mention the untimely demise you've perpetrated upon the counting numbers beyond 4, which must now pine away forsaken.

And after that much is done, there remains the task of matching up all of the irrationals.

As there are ways to match the integers with the rationals without creating ad hoc redefinitions, and you haven't managed even that much — note there are algebraic numbers (roots of polynomials over the rationals) in the queue as well — it's time to give up, and just perhaps acknowledge that mathematicians do know what we're doing. Cantor's proof does actually work using the definitions we use, and if it no longer works after you've changed the definitions, that doesn't make Cantor an idiot or create leave for you to speak of his work as nonsense.

That last is really annoying. He's dead. Don't be taking cheap shots at him.

There are reasons why we use the definitions we do, and why those definitions are fixed. For one thing, it gives scads of us, working together, the chance to one-up each other by finding the errors in each others arguments, arguments that would never arise were we redefining common terms in the middle of a discussion.

On your path lies mayhem and madness! Turn back, I say, turn back! Think of the children!

This is why the set of Integers is infinite (i.e. endless).

This process clearly has a predecessor as you can always count up to where your dots begin.

And the process clearly always as a successor because you can always continue counting beyond any point where you have temporarily stopped.

This is precisely why the set of Integers is infinite to being with.

We already have an infinite set. There is no reason to pretend that we need a larger infinity.

Infinity = endless. Period.

There is no such thing as larger and smaller infinities. That's Cantor's nonsense.

In regards to whatever redefinition of infinity you might have in mind, please be advised that we already own that word, and have owned it since before you were born. It is not your word. Go find one of your own.

But more directly, we need larger infinities because there are sets we encounter naturally in our investigations with members which cannot be encompassed merely by counting up to infinity once. They require us to to do so again and again, an infinite number of times, in order to encounter all of their members. Their members can't be counted. This surprising result should be noted, and to do so with elegance, there are the uncountably infinite cardinals.

As ever, Jesse