Why Cantor's Diagonalization Proof is Flawed.

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micatala
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Post by micatala »

I know, I know, I said I was done with this . . .

I think the crux of Divine Insight's error is that he/she continues to insist that the number of nonzero digits must dictate the representation be rectangular. This is an invalid assumption.

There is no valid reason one cannot simply add zeroes to DI's 'rectangular' representations and still use the same set of actual numbers.

Another problem invaliding DI's argument is that DI refuses to acknowledge something as basic as 0.333 . . . =1/3 or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics.

He gives no valid reason for this rejection. He simply declares such notions as absurd or uses similar subjective or vague language to dismiss this.


And the further problem is that he assumes things at the finite level must be equally valid at the infinite level. Again, this is an invalid assumption. I believe, for example, he refused to address that an infinite set can have propers subsets of the same cardinality while this cannot be true of finite sets.



Divine Insight wrote: 1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list.
No you have not. You have simply insisted repeatedly that your representation is the only valid one by fiat.

You continue to have an ambiguous definition of 'complete' which it seems applies only to the representation, the numerals, and not the actual numbers. When I changed the representation, you insisted I had to change the set. Again, no valid basis for this claim.

00.00
01.00
10.00
11.00

is a complete list of the set (0, 1, 2, 3} expressed in binary notation. Complete, as understood in Cantor's proof, means the entirety of a set assumed to be countably infinite. If DI wants to say complete should apply to other types of sets, that is possible, but the term should mean 'the entirety of a given set' in some sense, or it has nothing at all to do with Cantor's proof, which would mean, as I showed above, DI's argument is a straw man.

If DI wants to insist instead that 'complete' means all possible numbers of a given number of digits using a particular representation, that could be used, but then his argument is irrelevant to Cantor's argument.

If we used base two representations instead of base 10 representations in Cantor's proof, the number of nonzero digits in some of the numbers in the list (whichever ones had a finite number of nonzero digits) would change, but the argument would be the same, just using only 0 and 1 as digits.




2. I have also demonstrated that it is the nature of these numerical lists to grow exponentially in the number of rows required to record every numeral with every column appended.

Do you disagree with this? If so place address that point.
Your rows versus column claim only is valid if you stick to your invalid assumption that your rectangular representation scheme is the only one. If you stick to this assumption, then you are dealing with representations as the key concept, not actual sets of numbers. Any of your finite representations could be made to be square by adding a finite number of zeroes. You refuse to acknowledge this, without basis in logic or fact. It does not matter if the different representation could represent more numbers at the finite level. The different representation still represents the same set original set of numbers. The new number created by going down the diagonal will still not be in the original set.

But really, the big problem is DI's refusal to accept the logical validity of infinite decimal representations.
3. I have shown that crossing off digits using a diagonal line that crosses off the next digit in the next row necessarily creates a square list of crossed off numerals. Therefore that list cannot be a complete list of numerals, nor can it be used to prove what may or may not be on a complete list of numerals.

Do you disagree with this? Is so please explain why without resorting to the absurd claim that when taken to infinity a complete list of numerals will magically become square. Where is there any rationale for such an absurd claim? I have already shown that completed lists must necessarily grow exponentially in rows with every digit column added. Therefore there is no justification for claiming that these innately rectangular lists will magically become square at infinity. The situation only gets worse with every digit crossed off the list. What's going to magically make the situation get better as you approach infinity?
DI labels a claim as absurd without any actual valid refutation, and the claim itself continues the 'squareness' straw man argument..

The claim that a 'square list of crossed off numbers' is created when you cross off a single number in each row is at best unclear if not nonsensical.

And again, 'square' is a property of the representation, not the set of numbers itself. It is an invalid concept with respect to Cantor's proof which does depend on possibly infinite decimal representations but does not depend on this vague notion of 'squareness' which would not even seem to make sense given DI is applying it to an 'infinite by infinite' square.
You need to explain how that works. You can't just claim that it will magically become square at infinity. Mathematics doesn't stand on unsubstantiated claims, or does it?
DI creates the ambiguous notion of squareness at infinity and then berates mathematicians for 'not understanding' his ad hoc notion. Neither Cantor nor any other mathematicians I am aware of makes claims about 'squareness at infinity' in this context. This is DI's straw man rearing its head again.


Therefore I have successfully demonstrated why Cantor's diagonal proof does not prove what he claims.
DI has not demonstrated this, but seems unwilling or incapable of accepting that.



Since DI's argument hinges on insisting without a valid reason that a representation like

00.00
01.00
10.00
11.00

must be considered invalid in favor of his rectangular representation, even though both represent the same set, his argument is an invalid straw man.

Consider, for example, that while Cantor's proof typically is phrased in terms of the real numbers between 0 and 1, it could just as easily been phrased in terms of the real numbers between N=10^(1000^1000) and N+1. We just add 1 and the appropriate number of zeroes ahead of the decimal point. This would make, in the same vague sense that DI insists Cantor's list be square, the representation rectangular but wider than it is long by the number of digits in N.
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Post by mgb »

DivineInsight wrote:1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list.
It doesn't. I have already shown this. All one has to do is write the zeros:

0.2000000000000000000000000000000000000000000000000000000000000...so it can be made square.
2. I have also demonstrated that it is the nature of these numerical lists to grow exponentially in the number of rows required to record every numeral with every column appended.
They don't have to grow like this. The list can be made square by adding zeros. You are making much ado over mere notation. Adding zeros allows the diagonal to be drawn because the array is 'square'.
If we replace all these zeroes with 1's then we end up with 0.111 which we claim is not on "our" list. That's fine, the only problem is that it's obviously already on the list at row number
You are using numbers with a finite number of digits. Cantor's numbers have an infinity of digits and this fact allows him to create a new number. You cannot apply finite arguments to infinities.

This is the list you gave:
0.000
0.100
0.010
0.110
0.011
0.100
0.101
0.111

and your diagonal is 111.

This is what it should look like:

0.000000000000000000000000000000000000000000...
0.100000000000000000000000000000000000000000...
0.010000000000000000000000000000000000000000...
0.110000000000000000000000000000000000000000...
0.011000000000000000000000000000000000000000...
0.100000000000000000000000000000000000000000...
0.101000000000000000000000000000000000000000...
0.111000000000000000000000000000000000000000...

where there are an infinity of zeros to make it 'square'. Now take the diagonal and add 1 to each digit. Your diagonal is now:
11111111...
not 111

DivineInsight wrote:You are still ignoring the fact that with every digit you add you need to add 10^n row if you want to claim that your list is complete.
That does not matter if I am adding an infinity of zeros. 10^n x infinity = infinity.
Since when has mathematics become a game of just making unsubstantiated claims as you are doing?
It is not an unsubstantiated claim. It only involves the simplest rules of notation.

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Post by Divine Insight »

micatala wrote: I know, I know, I said I was done with this . . .

I think the crux of Divine Insight's error is that he/she continues to insist that the number of nonzero digits must dictate the representation be rectangular. This is an invalid assumption.

There is no valid reason one cannot simply add zeroes to DI's 'rectangular' representations and still use the same set of actual numbers.
Yes there is a valid reason. You cannot simply add zeroes in an attempt to make a rectangular list square. Every zero you add demands that your list must necessarily become exponentially more rectangular. So adding zeroes while pretending to ignore the innate property of numerical lists is not valid.

micatala wrote: Another problem invaliding DI's argument is that DI refuses to acknowledge something as basic as 0.333 . . . =1/3 or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics.
First off, both of these accusations are false. I never refused to acknolwedge that 0.333.... = 1/3 in the calculus limit. To the contrary I acknowledge this. But anyone who understands calculus knows that this does not mean that 0.333... = 1/3 in any other way.

So you are misrepresenting my position on that, plus this is a total distraction from the problem at hand which you clearly do not yet understand since you still think you can just add zeros without affecting the length of the list.
micatala wrote: ... or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics.
Exactly. Whether an irrational relationship should be defined as a real number is a totally different topic and has absolutely nothing at all to do with the argument Cantor is making.

So to even bring this up is an off-topic strawman that has nothing at all to do with the arguments put forth in this thread. Whether the square root of 2 should be defined as a real number or not has nothing to do with the problem at hand.

So clearly you are grasping at straws because you obviously cannot address the points I've actually made.
micatala wrote: And the further problem is that he assumes things at the finite level must be equally valid at the infinite level. Again, this is an invalid assumption. I believe, for example, he refused to address that an infinite set can have propers subsets of the same cardinality while this cannot be true of finite sets.
Set theory has nothing at all to do with this topic. Cantor's argument is based on listing numerals and crossing off digits claiming that he has created a number that isn't on the list. But I have shown that this is a false claim on Cantor's part. Set theory doesn't even need to be mentioned at all.

So again, you haven't address the points I've made. All you've done is try to distract away from them.
micatala wrote:
Divine Insight wrote:

1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list.

No you have not. You have simply insisted repeatedly that your representation is the only valid one by fiat.

You continue to have an ambiguous definition of 'complete' which it seems applies only to the representation, the numerals, and not the actual numbers. When I changed the representation, you insisted I had to change the set. Again, no valid basis for this claim.

00.00
01.00
10.00
11.00
There is nothing ambiguous about my definition of 'complete'. Apparently you simply haven't yet understood the argument. The list you have listed above is far from complete.

Look at this list:

00
01
10
11

That is a complete list of all numbers that can be represented by a 2-digit binary representation. There no other possibilities. This is a complete list.

Your list is far from complete. I can name 12 more valid numbers that don't appear on your list. In fact, any number you create by drawing a diagonal line down your list and replacing 0's or 1's with their opposite will necessarily be one of the 12 numbers that I can name that belong on that list if that list is said to be complete. Necessarily so.

So once again, you've only demonstrated that you don't understand my first statement.

Your list is incomplete, mine is not.
micatala wrote: If DI wants to insist instead that 'complete' means all possible numbers of a given number of digits using a particular representation, that could be used, but then his argument is irrelevant to Cantor's argument.
Hardly. Cantor's argument is an argument where he claims to be constructing a number that isn't on the list. But he can never do that because his list is necessarily square and therefore an invalid list.
micatala wrote: If we used base two representations instead of base 10 representations in Cantor's proof, the number of nonzero digits in some of the numbers in the list (whichever ones had a finite number of nonzero digits) would change, but the argument would be the same, just using only 0 and 1 as digits.
Exactly. And not only would it be the same for the so called "real numbers" but it would also be the same for even the natural numbers. So this even shows more vividly that Cantor's argument is meaningless.
micatala wrote: Your rows versus column claim only is valid if you stick to your invalid assumption that your rectangular representation scheme is the only one. If you stick to this assumption, then you are dealing with representations as the key concept, not actual sets of numbers.
Well, if we get two different answers using actual numbers versus sets, then clearly set theory has its own problems. And that most certainly wouldn't surprise me in the least. If fact, mathematicians already know that set theory has problems so there's nothing new there.
micatala wrote: DI labels a claim as absurd without any actual valid refutation, and the claim itself continues the 'squareness' straw man argument..

The claim that a 'square list of crossed off numbers' is created when you cross off a single number in each row is at best unclear if not nonsensical.
It's neither unclear nor nonsensical.

The list Cantor is creating is created by crossing off 1 digit per row. Therefore his resulting list can never be anything other than square. And far more importantly I have already shown that every single number he creates using this method must necessarily already be included on any complete list take out to that many columns.

But I will grant you this much. Cantor's error was indeed due to his concentration on trying to create sets. And this is no doubt why he totally missed the fact that no complete list of any numerical system can possible be square.

By looking at it from a pure set theoretic vantage point he couldn't see the forest for the trees.

This actually shows how focusing on set theory can get us into deep trouble that we wouldn't have otherwise found ourselves in.
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Post by mgb »

micatala wrote:DI creates the ambiguous notion of squareness at infinity and then berates mathematicians for 'not understanding' his ad hoc notion. Neither Cantor nor any other mathematicians I am aware of makes claims about 'squareness at infinity' in this context. This is DI's straw man rearing its head again.
Yes it is purely a question of notation. The list can easily be made square by adding an infinity of zeros making it as wide as it is long. Just write 1.00 as

1.00000000000000000000000000000000000000000000000...

Edit: I just read the quote and see you have already made this very point.
Last edited by mgb on Tue Apr 23, 2019 1:23 pm, edited 1 time in total.

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Post by mgb »

DivineInsight wrote:Yes there is a valid reason. You cannot simply add zeroes in an attempt to make a rectangular list square. Every zero you add demands that your list must necessarily become exponentially more rectangular. So adding zeroes while pretending to ignore the innate property of numerical lists is not valid.
That is irrelevant. As I said, the list cannot be 10^n times higher than wide IF IT IS INFINITELY WIDE.

Also, there is no need to make the list in ascending order, writing the smallest numbers first. Cantor's list is randomly shuffled and his first number has an infinity of digits...

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Post by micatala »

Divine Insight wrote:
Divine Insight wrote:
micatala wrote: I know, I know, I said I was done with this . . .

I think the crux of Divine Insight's error is that he/she continues to insist that the number of nonzero digits must dictate the representation be rectangular. This is an invalid assumption.



There is no valid reason one cannot simply add zeroes to DI's 'rectangular' representations and still use the same set of actual numbers.

Yes there is a valid reason. You cannot simply add zeroes in an attempt to make a rectangular list square. Every zero you add demands that your list must necessarily become exponentially more rectangular. So adding zeroes while pretending to ignore the innate property of numerical lists is not valid.


What rule of numbers or mathematics says that I cannot add zeroes at the end of a given number? It's the same number with or without the zeroes. You say "cannot." I see no valid reason why not. The fact that adding more digits via zeroes allows additional numbers to be represented is not a valid reason. With or without the additional zeroes, and whether or not the representations is 'square,' neither the individual numbers nor the set of numbers changes.

Changing the representation does not change the underlying function between the sets.

00 -> 0
01 -> 1
10 -> 2
11 -> 3

is the same exact bijective function between the exact same two sets as

00.00 -> 0
01.00 -> 1
10.00 -> 2
11.00 -> 3

It seems to me, you continue to be confused by the ambiguous way in which you are using the term 'complete,' although I accept you can define it as you clarified below.
micatala wrote: Another problem invaliding DI's argument is that DI refuses to acknowledge something as basic as 0.333 . . . =1/3 or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics.
First off, both of these accusations are false. I never refused to acknolwedge that 0.333.... = 1/3 in the calculus limit. To the contrary I acknowledge this. But anyone who understands calculus knows that this does not mean that 0.333... = 1/3 in any other way.

So you are misrepresenting my position on that, plus this is a total distraction from the problem at hand which you clearly do not yet understand since you still think you can just add zeros without affecting the length of the list.
Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333. . . Labeling the latter as a 'calculus limit' in order to dismiss the legitimate use of the infinite decimal expansion is simply a semantic game without basis in logic.

Again, taking your clarification below into account, you still have an error in applying the finite situation to the infinite.

OK, so let's say adding the zeroes to create a finite square now means I have to make the list longer to include all possible numbers that can be represented with that many zeroes, both zero and nonzero after the decimal point, in order to make the list 'complete' in the sense you understand it.

The problem is your argument is irrelevant when you move to the case Cantor is actually considering.

The infinite list that Cantor has cannot get longer in any meaningful way. This is due to the previous point I made about infinite sets having subsets of the same cardinality.

If I add one number to Cantor's list, say the one he creates through diagonalization, the list is not really any longer. As an illustration, consider the following function.

1 -> 11
2 -> 12
3 -> 13
etc.

In one sense, the domain of this functions, all natural numbers is 'longer' having 10 elements that are not in the range (the set of natural numbers greater than 10). But the one-to-one correspondence shows they are the same cardinality, which I suppose you could say means they are the same length. Thus, adding any number of numbers to an infinite set, even adding countably many, does not change the length of the list in any meaningful way.

In fact, I already noted the function f(n)=2n gives a bijection from the natural numbers to the set of even numbers.

The function f(n)=2^n gives a bijection between the natural numbers and the powers of 2, even though you could say the powers of 2 represent in an intuitive sense an 'exponentially smaller' or 'shorter' set.







micatala wrote: And the further problem is that he assumes things at the finite level must be equally valid at the infinite level. Again, this is an invalid assumption. I believe, for example, he refused to address that an infinite set can have propers subsets of the same cardinality while this cannot be true of finite sets.
Set theory has nothing at all to do with this topic.
What????

Set theory is part and parcel of this whole question. The whole point is to prove the set of real numbers has a different cardinality than the set of natural numbers. Cardinality is a concept in SET THEORY. It is defined using bijective functions between sets.

Cantor's argument is based on listing numerals and crossing off digits claiming that he has created a number that isn't on the list. But I have shown that this is a false claim on Cantor's part. Set theory doesn't even need to be mentioned at all.
Cantor's argument is based on an assumed (for purposes of contradiction) bijective function between sets. The argument depends on the fact that real numbers have decimal expansion representations, and many (most!) numbers have infinite decimal representations. The bijective function can be considered even without considering how the real numbers might be represented.

Again, the main problem is your insistence without any logical basis in dismissing infinite decimal representations.


DI wrote:
micatala wrote:
You continue to have an ambiguous definition of 'complete' which it seems applies only to the representation, the numerals, and not the actual numbers. When I changed the representation, you insisted I had to change the set. Again, no valid basis for this claim.

00.00
01.00
10.00
11.00
There is nothing ambiguous about my definition of 'complete'. Apparently you simply haven't yet understood the argument. The list you have listed above is far from complete.

Look at this list:

00
01
10
11

That is a complete list of all numbers that can be represented by a 2-digit binary representation. There no other possibilities. This is a complete list.

Your list is far from complete. I can name 12 more valid numbers that don't appear on your list. In fact, any number you create by drawing a diagonal line down your list and replacing 0's or 1's with their opposite will necessarily be one of the 12 numbers that I can name that belong on that list if that list is said to be complete. Necessarily so.

So once again, you've only demonstrated that you don't understand my first statement.

OK. I will accept you have now clarified that your use of the word 'complete' corresponds to my second description. Using your now clarified definition, my list is not complete. However, the question is whether your notion of complete is relevant to Cantor's proof.
DI wrote:
micatala wrote: If DI wants to insist instead that 'complete' means all possible numbers of a given number of digits using a particular representation, that could be used, but then his argument is irrelevant to Cantor's argument.
Hardly. Cantor's argument is an argument where he claims to be constructing a number that isn't on the list. But he can never do that because his list is necessarily square and therefore an invalid list.
Not being square does not make the list 'invalid.' I see no valid reason for this claim. Secondly, speaking of a list being square or not makes no sense with respect to infinite lists.

Here is an infinite list of integers.

1
11
111
1111
etc.

I suppose you could in an intuitive sense call this list 'triangular.' But it could just as easily be made 'square' by putting in a decimal point and infinitely many zeroes afterwards for each number. 'Squareness' simply has no definite meaning with infinite lists. Even if it did, a list not being 'square' is irrelevant to Cantor's argument since each number is represented by an infinite decimal expansion.

And there is no valid reason for dismissing infinite decimal expansions, whether you use the term 'calculus limit' or not.


micatala wrote: Your rows versus column claim only is valid if you stick to your invalid assumption that your rectangular representation scheme is the only one. If you stick to this assumption, then you are dealing with representations as the key concept, not actual sets of numbers.
Well, if we get two different answers using actual numbers versus sets, then clearly set theory has its own problems. And that most certainly wouldn't surprise me in the least. If fact, mathematicians already know that set theory has problems so there's nothing new there.
Your response is a nonsequitur. The sets in question have actual numbers as their elements. Cantor's argument involves a bijective function between sets of numbers. I have no idea what 'two different answers' you are accusing me of here. Two different representations of the same set still represent one set.

micatala wrote: DI labels a claim as absurd without any actual valid refutation, and the claim itself continues the 'squareness' straw man argument..

The claim that a 'square list of crossed off numbers' is created when you cross off a single number in each row is at best unclear if not nonsensical.
It's neither unclear nor nonsensical.

The list Cantor is creating is created by crossing off 1 digit per row. Therefore his resulting list can never be anything other than square. And far more importantly I have already shown that every single number he creates using this method must necessarily already be included on any complete list take out to that many columns.
You have several errors here. Cantor does not 'cross digits off.' He constructs a number by selecting digit N to be different than digit N in the Nth number in the assumed (for purpose of contradiction) bijective function. He says nothing about his process resulting in a 'square.' The 'squareness' of the finite representation of his infinite list is irrelevant and nothing more than a vague intuitive notion.

Your claim about the number he creates already being on the list is faulty reasoning based on your invalid assumption that what works for finite lists of numbers with finitely many digits must necessarily apply to what Cantor is doing with an infinite list of numbers with infinitely many digits.

But I will grant you this much. Cantor's error was indeed due to his concentration on trying to create sets. And this is no doubt why he totally missed the fact that no complete list of any numerical system can possible be square.
This again is an error on your part as what is true for finite lists of numbers with finitely many digits need not be true for infinite lists.



You really should go read the Mathematical Cranks book. It covers a lot of discussions like the one we are having. It documents how even a well-meaning amateur can go off the rails by continuing to insist that their thinking must be right while all the mathematicians must be wrong. Many of these examples exhibit the type of errors in thinking you are exhibiting here. Again, I hate to be unking, but that is the reality of this situation.
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Post by Divine Insight »

micatala wrote: Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333.
It has no exact decimal representation. You can obviously use decimal notion to approximate 1/3 as close as you like. However since you can never write out an infinite number of 3's or actually use that representation in any real way.

What good does it do to even claim that 0.333... equals 1/3? Other than a useless empty claim it has no practical value. And even in mathematics, when we say that 0.333... = 3 we're actually making this statement using the calculus limit as our justification. So why not just say, "In the calculus limit 0.333... = 3". To do anything less is nothing short of laziness. What you are arguing for is nothing short of lazy mathematics.
micatala wrote: What rule of numbers or mathematics says that I cannot add zeroes at the end of a given number?
Obviously it depends on context. If you are claiming to have complete list of numbers for that numerical system then you can't just arbitrarily add zeroes without taking into consideration how that will affect the complete list.
micatala wrote: It seems to me, you continue to be confused by the ambiguous way in which you are using the term 'complete,' although I accept you can define it as you clarified below.
I've clarified this several times over already. By 'complete' I simply mean every number that a numerical system can represent.

You seem to be ignoring the context of Cantor's argument. Cantor is trying to claim to be able to produce a number that isn't on the list. Well, if his list isn't a 'complete' list then his claim to have created a number that isn't on his incomplete list is utterly meaningless. He hasn't proven anything if the list he's creating is incomplete.

So that's why it's paramount in this situation and cannot be ignored.
micatala wrote: Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333. . . Labeling the latter as a 'calculus limit' in order to dismiss the legitimate use of the infinite decimal expansion is simply a semantic game without basis in logic.
Two things:

First off, even in mathematics to say that 0.333... = 3 is nothing more than a semantic game. Give me an concrete example where this could even be used in a practical real live situation. It can't because it's meaningless to even speak of 0.333... existing in any real world sense.

So you all you are doing is arguing for a mathematics that is itself nothing more than semantic games.

Secondly, this question has absolutely nothing at all to do with the question of Cantor's diagonal proof. Even if 0.333... does equal 1/3 exactly that still doesn't help Cantor's problem.
micatala wrote: Set theory is part and parcel of this whole question. The whole point is to prove the set of real numbers has a different cardinality than the set of natural numbers. Cardinality is a concept in SET THEORY. It is defined using bijective functions between sets.
But Cantor isn't making a bijection between sets. He's attempting to make a bijection between the set of natural numbers AND a numerical representation of decimals.

Not only this, but if he is going to claim this his "list" represents the set of decimals then we can see that his set is incomplete. Precisely because his list of decimals is necessarily incomplete because it's necessarily a square list. Therefore there should be no surprise at all that he arrives at the absurd conclusion that he is creating numbers that aren't on the list. Of course he would. That's exactly what his method much do.

So it's a false representation of a bijection of sets. His list is not the same as the actual set of decimal numbers.

Therefore if he's trying to make a bijection between sets his method fails miserably as I have been describing.
micatala wrote: Cantor's argument is based on an assumed (for purposes of contradiction) bijective function between sets. The argument depends on the fact that real numbers have decimal expansion representations, and many (most!) numbers have infinite decimal representations. The bijective function can be considered even without considering how the real numbers might be represented.
I agree. I'm not renouncing the bijective function. A bijective correspondence can indeed be made between two sets. However, in this case that's not what Cantor is doing. In this case his representation of the set of decimal expansions is incomplete. Therefore it's not a valid bijection between sets.
micatala wrote: Again, the main problem is your insistence without any logical basis in dismissing infinite decimal representations.
This is an incorrect statement of my position. I have never dismissed infinite decimal expansions. All I'm saying is the list that Cantor is construction is not a complete list of decimal expansions. Period.

So when he claims to be creating numbers that aren't on his list it's a meaningless claim. Just because he is creating numbers that aren't on his incomplete list does not mean that those numbers do not exist in the actual set of decimal expansions.

He's claiming to be creating numbers that aren't on the list. And that's his error.
micatala wrote: OK. I will accept you have now clarified that your use of the word 'complete' corresponds to my second description. Using your now clarified definition, my list is not complete. However, the question is whether your notion of complete is relevant to Cantor's proof.
Finally some progress. :D

Cantor's entire proof requires that "his list" contains all possible decimal expansions. Why? Because he's claiming to be producing a number that cannot be on the list. Therefore if his list does not contain all possible decimal expansions then his proof is meaningless. The fact that he can create numbers that aren't on 'his list' says nothing about the actual set of decimal expansions.

So his proof fails to show what he had hoped to be showing.

Apparently he didn't catch this, because he just assumed that lists of decimals could be listed in a square list.

Why does his list have to be square? Because he's using a diagonal line to create then numbers he's claiming are not on his list. But that method doesn't work for rectangular lists. Why? Because his diagonal cross-out-line simply can't descend down the list at a fast enough rate to cover all the possible decimal expansions. This is an innate property of the numerical representation of numbers that Cantor apparently never even gave a thought to. He was so concerned with making an abstract bijection between sets that he failed to realize that his diagonal line construction cannot be used to make this bijection.
micatala wrote: 'Squareness' simply has no definite meaning with infinite lists. Even if it did, a list not being 'square' is irrelevant to Cantor's argument since each number is represented by an infinite decimal expansion.
You seem to be too hung up on the geometric idea of a square. Perhaps it's that term that is confusing you. The point is that any complete list of numbers expressed using numerical symbols must necessarily have more rows than columns. Ignoring this fact in an attempt to imagine a geometrically square list does nothing to address the core issue.

The fact that Cantor can produce numerical representations of numbers that aren't on his list is meaningless, because his diagonal line method cannot cross off more than one digit per row. And that's simply not enough to perform the task he needs.

micatala wrote: Your response is a nonsequitur. The sets in question have actual numbers as their elements. Cantor's argument involves a bijective function between sets of numbers. I have no idea what 'two different answers' you are accusing me of here. Two different representations of the same set still represent one set.
But Cantor's list of decimal representations cannot be valid representation of the set of all real numbers.

Let's not forget that he is claiming to be creating numbers that aren't on the list (i.e. aren't in the set real numbers. Talk about something that is nonsequitur.
micatala wrote: You have several errors here. Cantor does not 'cross digits off.' He constructs a number by selecting digit N to be different than digit N in the Nth number in the assumed (for purpose of contradiction) bijective function. He says nothing about his process resulting in a 'square.' The 'squareness' of the finite representation of his infinite list is irrelevant and nothing more than a vague intuitive notion.
He does indeed 'cross digits off' why do you think they call this the diagonal line proof?

He constructs his number by crossing off digits and replacing them by his free choice of other numerals. And what other numerals is he choosing? He's choosing from numerals that make up the very notation that he is using. But he hasn't taken into account that those numerals require more rows than columns to fully describe these numbers.

He doesn't need to mention anything about "squareness". In fact, it's pretty clear that he ever even realized this folly. The squareness of his construction comes from the fact that he can only cross off one digit per row. So he's stuck with that whether he likes it or not.

And of course it follows that he'll create numbers that aren't on the list. That's naturally going to happen in any case because complete numerical lists of numbers necessarily require more rows than columns. So if you go down any complete list of numerals constructing a new number using a diagonal line that crosses off 1 digit per row you will ALWAYS produce a number that isn't on the list above where you are currently working.

In other words, this method of creating new arrangements of numerals is innate to the process he's invented. It has nothing at all to do with the actual properties of any sets. Yet he's claiming to have made a valid bijection between sets. But he hasn't done that at all. All he has managed to do is ignore the innate property of complete lists of numerals.

micatala wrote: Your claim about the number he creates already being on the list is faulty reasoning based on your invalid assumption that what works for finite lists of numbers with finitely many digits must necessarily apply to what Cantor is doing with an infinite list of numbers with infinitely many digits.
Pretending that this process could be legitimately taken to infinity is a grave error.

I have already shown how and why Cantor's list get's further and further behind the real list of decimals with every digit he crosses off.

If anything he's getting infinitely behind the set of reals.

How in the world can you expect to take a process out to infinity when the process gets further and further behind with ever step you take?

In order to show that something works at infinity you need to show that it works for every step on the way toward infinity and there is nothing that would prevent it from continuing.

You can't just claim that a process that can't even be made to work at all will somehow magically correct itself at infinity. Where is there any justification for that?

I showed why this thing can't even get off the ground, but you want to take it to the edge of the universe and claim that by the time it gets there everything will magically be ok.

Where's the justification for that?

If you can't even get it off the ground you aren't going to make any journey to infinity.
micatala wrote: This again is an error on your part as what is true for finite lists of numbers with finitely many digits need not be true for infinite lists.
It's hardly a proof of anything if the only thing you can claim about it is that it will magically repair itself at infinity if we simply ignore the fact that it doesn't even work at square one.
micatala wrote: You really should go read the Mathematical Cranks book. It covers a lot of discussions like the one we are having. It documents how even a well-meaning amateur can go off the rails by continuing to insist that their thinking must be right while all the mathematicians must be wrong. Many of these examples exhibit the type of errors in thinking you are exhibiting here. Again, I hate to be unking, but that is the reality of this situation.
Sorry, but even the most prominent mathematicians fully realize that there are major problems with our mathematical formalism. They discuss these issues all the time.

In fact, mathematicians can't even agree on whether our mathematical formalism is invented or discovered.

Here's a trailer of a recent 3-part documentary on precisely this topic.

[youtube]https://www.youtube.com/watch?v=lMaoAWB03RQ[/youtube]

If you think that mathematics is fully understood, or even truly trustworthy in all of its claims, you really need to rethink that.

I've already answered many of the questions that Hanna Fry asks in this video. And it's also true that many mathematicians don't even agree on many of the things that the mathematical community has chosen to include in the formalism.

So if you think that mathematicians are all in agreement on these things, then you need to stop reading books about cranks and start reading books about real mathematicians. You'll quickly find that they have vast disagreements on many mathematical issues.

Mathematics if far some being carved in stone. And mathematics, being human, are indeed prone to making mistakes.

Our ideas about abstract concepts such as infinity are indeed quite abstract, and there is no reason to accept that these ideas are anything more than the imagination of men.

In fact, I'm not the only one who questions Cantor's ideas. There are actually quite a few mathematicians that don't take his ideas seriously. And rightfully so. Cantor treats infinity as though it is a finite quantity. How utterly absurd is that?
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mgb
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Post by mgb »

Your objection is based purely on a misunderstanding of decimal notation. Those zeros exist implicitly in the notation. They are as valid as any non zero digit in a real number representation. We don't normally write the zeros because it is not practical but Cantor's method requires a perfect representation of real numbers and this representation requires the zeros. It has been repeatedly pointed out to you, by the simplest arguments, that you are wrong and you refuse to accept it.

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Post by Divine Insight »

mgb wrote: Your objection is based purely on a misunderstanding of decimal notation. Those zeros exist implicitly in the notation. They are as valid as any non zero digit in a real number representation. We don't normally write the zeros because it is not practical but Cantor's method requires a perfect representation of real numbers and this representation requires the zeros. It has been repeatedly pointed out to you, by the simplest arguments, that you are wrong and you refuse to accept it.

Sorry, but your argument doesn't hold in Cantor's situation.

Why? Because Cantor is changing the zeroes to 1's in order to create his new number that he claims isn't on his list.

So his argument cannot treat those trailing zeros as being nothing more than place holders. If he's going to be changing them to 1's then he can't use your argument.

You can't change those zeroes to 1's and claim that it doesn't make any difference.

In fact, in Cantor's actual argument, since he's using base ten, he allows that those zeroes can be replaced with any digit between 1 and 9. So he's giving that decimal position the full potential.

So he cannot use your argument to defend his diagonal line argument.

You are thinking in terms of representing just one number. Cantor is talking about creating an entire and complete list of numbers. It's important that Cantor's list be a complete list. Otherwise it would be meaningless for him to claim that he is creating new numbers that aren't on his list.

After all, if his list is incomplete, then what would it matter if he could create a number that isn't on the list? That would already be a given.

If you are creating an incomplete list of numbers and you show that there are numbers that cannot be on your incomplete list why should anyone care about that?

It's paramount that Cantor's list be a complete list of real numbers. Otherwise, claiming to have created a number that isn't on the list would be meaningless.

I have shown why the list that Cantor is creating cannot be a complete list of real numbers.

Keep in mind that his entire proof is based on the claim that his list (i.e. the list he has already created by drawing his diagonal line) cannot contain the new numbers he is creating. But his list is necessarily square (i.e. contains the same number of rows and columns). This is necessarily so because that's the only kind of list he can create using a diagonal line that creates one new row per digit.

Therefore the list Cantor is creating cannot be a complete list of real decimals.

Thus his so-called "proof" fails.

So if anyone wants to claim that the natural numbers cannot be put into a 1-to-1 bijection with the real numbers they'll need to use some other method to show that this is the case because Cantor's diagonal line method doesn't cut it.

Cantor's diagonal line proof fails, and mathematicians should have recognized this by now. Shame on them.

Note: I'm not saying that the conclusion of Cantor's argument is necessarily false. He might have lucked out and the conclusion just happens to be true for some other reasons. I'm not trying to claim that the conclusion is false. All I'm saying is that Cantor's diagonal line proof fails to prove his conclusion. So mathematicians need to have this taken out of textbooks because it's a bogus proof.

If they can find some other way to show that the real numbers cannot be put into a bijection with the natural numbers more power to them. But Cantor's diagonal line proof fails to establish this to be a fact. And mathematicians need to wake up and recognize this fact.

I've just explained why it fails. So they have no excuse for not acting on this information and having Cantor's proof rejected as being a flawed proof.

They'll just have to live without this specific proof. If they can show why the natural numbers cannot be placed in a 1-to-1 correspondence with the real numbers using some other credible proof, then they can keep that conclusion. But if Cantor's diagonal line proof is all they have, then they'll even need to retract that conclusion as having been proven.
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