# A structural point of view on Cantor's diagonal arguments

Organic
Hi,

Thank you.

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The main idea of Cnator's second argument is to show that the real numbers list can never be a complete list.

If we have to correct the list by adding to it infinitely many Cantor's function results, it means that there is no bijection between |R| and |N|.

Cantor's first argument clearly shows that there is a bijection between |N| and |Q|.

There is no problem to represent any rational number by its decimal form.

And Q numbers decimal's form is finite or it is infinitely many digits with repetitions over scales.

But when we use Cantor's second argument on the decimal representations of Q numbers, we find exactly the same results as Cantor found between |N| and |R|.

|N|=|Q| by Cantor's first argument, but |N|<|Q| by Cantor's second argument.

My list is a decimal representation of any rational number in Cantor's first argument specific list.

For example:

0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...

In this case Cantor's function result is 0.0101010101010101... which is not in the list.

Every non-zero decimal digit can be any number between 1 to 9, Because I use Cantor's function where the rules are:

A) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.

B) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.

For example, there is no problem to use 0.333333... in the list because 3 (or any digit between 1 to 9) is always turned to 0, therefore our new number can always be a rational number.

After we add Cantor's function result to the list, we rearrange it in such a way, which gives us a new rational number as Cantor's function result, and so on and so on.

Is it ? NO it is not true, because Cantor's new number does not exist in our decimal list, but it exists in Cantor's first diagonal list. Therefore our decimal list is not complete.

But there is some interesting question that we can ask.

When we have a complete list of rational numbers, represented by their decimal form, then Cantor's function result cannot be but an irrational number.

I think we have here some interesting situation, because if what I wrote holds, it means that there can be some difference between aleph0 and aleph0-1, which is not quantitative but structural.

It means that if even one of the rational numbers is missing, we have the ability to define some rational number (repetitions over scales) as Cantor's function result.

But when we have a complete list of rational numbers, represented by their decimal forms, then Cantor's function result cannot be but an irrational number (no repetitions over scales).

Is there some mathematical area which deals with this fine difference between aleph0 and aleph0-1 ?

If we take the next step we can ask what is the structural difference between 2^aleph0 and 2^aleph0-1 ?

We know that Cantor's function can find some rational number as a result (repetitions over scales), only if our list is aleph0-1 (or "less").

Because we have only one representation to all irrational numbers, which is the base value expansion, how can we be sure that we are not in the same situation, which has been found in the case of the rational numbers list ?

In the case of the rational numbers we have two representation forms of numbers that can be compared to each other and help us to find the complete list.

But this is not the case of all R numbers that can be represented by only one form.

What I mean is when we still find some R number which is not in the list, can't we say that it means that we have 2^aleph0-1 list, which is not the 2^aleph0 complete list?

In this case we call |R| the uncountable, but maybe the only reason is the fact that we don’t know how to represent 2^aleph0 numbers, and all we have is the representation of 2^aleph0-1 numbers.

I know that 2^aleph0-1 = 2^aleph0, but again I am talking about the structural difference between |R| and |R|-1.

When the list has aleph0-1 numbers, then Cantor's function result can be some rational number (repetitions over scales).

When the list has aleph0 numbers, then Cantor's function result can't be but some irrational number (no repetitions over scales).

These are structural differences.

When the list has 2^aleph0-1 numbers, then Cantor's function result is some irrational number (no repetitions over scales).

When the list has 2^aleph0 numbers, then Cantor's function result is unknown because our base value expansion representation method is limited to two structural forms, which are repetitions, or no repetitions over scales.

Therefore, we can't conclude that |R| is uncountable, because Cantor's function has no input when our list has 2^aleph0 numbers.

Organic

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Homework Helper
But when we use Cantor's second argument on the decimal representations of Q numbers, we find exactly the same results as Cantor found between |N| and |R|.

No, we don't! That has been repeatedly explained to you. The "new" number that results is not rational.

My list is a decimal representation of any rational number in Cantor's first argument specific list.

For example:

0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...

In this case Cantor's function result is 0.0101010101010101... which is not in the list.

No, it isn't. You have not DEFINED your list completely. You have no way of knowing that the "new" number will be 0.0101010101...
In fact, it is IMPOSSIBLE to order the rational numbers in such a way.

Where, by the way, did you get that list? It doesn't look like the list used in any proof of the countability of the rationals that I have ever seen. That first number is 17113/9999 and that seems a very peculiar number to start with!

The form of the proof I am most familiar with starts the list:
1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, etc.

Organic
HallsofIvy,

Any way, you wrote:
In fact, it is IMPOSSIBLE to order the rational numbers in such a way.
By what proof?

I clime that it can't be done if and only if we have a complete list (aleph0 numbers) of Q numbers, represented by their base value expansion method.

If we have aleph0-1 Q numbers, then we still can rearrange the list in such a way, that gives us some rational number as Cantor's function result.

Please prove that I am wrong.

Organic

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Homework Helper
Yes, I read ALL of your post. No, I will not prove that you are wrong because you have been told repeatedly exactly why you are wrong. You have either not understood or have chosen to ignore what you were told.

You are the one who has a made a claim. You have claimed that your list contains ALL rational numbers but that the "diagonal method" produces a rational number that is not on that list. You have not defined the list precisely and so have no way of proving that the number .01010101... would be the number produced by the diagonal method. It is your responsibility to prove your claim, not the responsibility of others to disprove it.

Organic
OK, let us say that what I write is two theorems connected to each other.

I clime that if and only if Q decimal representation list is complete, then and only then Cantor's function result cannot be anything but an irrational number.

I also clime that when Q decimal representation list is NOT complete, then we still can rearrange the list in such a way that gives us a rational number as Cantor's function result.

I think these are legitimate climes that have to be answered.

If you think that they are not legitimate climes, please explain why ?

Thank you.

Organic

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Homework Helper
I will probably hate myself in the morning but, in the hopes that someone else reading this will get something out of it (and knowing that Organic will quite likely "thank" me for it and the quote some garbled version of to support another claim that Cantor didn't know what he was doing):

I clime that if and only if Q decimal representation list is complete, then and only then Cantor's function result cannot be anything but an irrational number.

I also clime that when Q decimal representation list is NOT complete, then we still can rearrange the list in such a way that gives us a rational number as Cantor's function result.

That was not what you claimed in your first post (in this thread).

You claimed 3 things:
1. That your list contains all rational numbers
My list is a decimal representation of any rational number in Cantor's first argument specific list.
2. That the number that "Cantor's diagonal process" produces, which is not on the list, is 0.0101010101...
In this case Cantor's function result is 0.0101010101010101... which is not in the list.
3. That 0.0101010101... is a rational number.
(You don't explicitely say so but that is the one statement that is obviously true.)

These can't all be true- they are contradictory!

If you think this is what Cantor did you are mistaken. Cantor said
1. Suppose THERE EXIST a list of all real numbers (i.e. suppose there is a one to one correspondence between N and R). Then
2. his "diagonal process" produces a number that is not on the list. (This is obviously true- it differs from each number on the list in at least one digit.) And
3. this number is a real number. Also obviously true- any number that can be written as a decimal is a real number.
Since these three statements are contradictory (as were yours) but it is obvious that 2 and 3 are true, statement 1, that THERE EXISTS such a list must be false- N and R do not have the same cardinality.
Notice that he did NOT assert that a specific list included all real numbers. If he had done that, he would just have showed that that assertion was wrong, which would tell us nothing about the cardinality of the real numbers.

In your case, since you have not uniquely defined your list of rational numbers it is impossible to tell whether it is 1 or 2 that is wrong but it is obvious that one of the must be.

Now you are saying that your claim is:
"if and only if Q decimal representation list is complete, then and only then Cantor's function result cannot be anything but an irrational number." I'm not sure I like the "if and only if" part but it is certainly true that if such a list is complete- if all rational numbers are on the list, then, since "Cantor's number" is not on the list it must be irrational!
The "only if" part of that, and your second claim:
"that when Q decimal representation list is NOT complete, then we still can rearrange the list in such a way that gives us a rational number as Cantor's function result." I'm not so sure about. Suppose we make a list of rational numbers that contains every rational number except 0.5 (which is certainly possible). Can we be certain that there exists an arrangement of the list so that "Cantor's diagonal process" gives 0.5 (which it would have to in order to give a rational number not on the list)? Cantor's process requires that we replace the nth digit in the nth by something other than that actual digit. Can we always rearrange the numbers so that the nth digit of the nth number is not 0? I suspect it may be true but you need to prove that, not just assert it. I also don't see what that has to do with what you were saying before.

Organic
HallsofIvy,

Thank you very much for your detailed post.

Rational and irrational numbers have different structures over scales, when you use their decimal representation.

Today math language can represent irrational numbers only by their decimal form.

If what I clime about aleph0 Q decimal list (Cantor’s function result cannot be anything but an irrational number) and aleph0-1 Q decimal list (Cantor’s function result still can be a rational number), then we can ask what about 2^aleph0 and 2^aleph0-1 list?.

What I want to say is this:

Cantor can’t say: ” Suppose THERE EXIST a list of all real numbers”.

And why he can’t say it, because we can represent this list only until 2^aleph0-1
different numbers, where the missing number is Cantor’s function result.

If we add this number to the list then the list is complete but then it no longer exist in any form, because in this case Cantor’s function result must has some structure which is not rational nor irrational.

If the diagonal cannot be represented in any known form, then the whole list cannot be represented, and we have nothing to conclude by Cantor’s method.

Organic

Homework Helper
Rational and irrational numbers have different structures over scales, when you use their decimal representation.
I have absolutely no idea what this means. What do you mean by "structures over scales"?

Today math language can represent irrational numbers only by their decimal form.
No that's not true. We can represent them in binary or octal or as &radic;(2) or &pi;. What exactly did you mean to say?

You have referred to aleph0 and aleph0-1.
There are many people who would say that aleph0-1 does not exist and as many who would say that aleph0-1= aleph0.
They are both right- they are talking about different mathematical structures. In either case, your point, which depends on there being a difference between aleph0-1 and aleph[0], is not correct.

Cantor can’t say: ” Suppose THERE EXIST a list of all real numbers”.
He can and did. One can say "suppose THERE EXIST" anything- and, perhaps, show that that leads to a contradiction.

If we add this number to the list then the list is complete but then it no longer exist in any form, because in this case Cantor’s function result must has some structure which is not rational nor irrational.
No, one can always add that number to the list but the list is still not complete because the same proof would show that there exist some number not in this new list. Cantor's proof actually shows that there are "uncountably many" numbers that are not on any such list.

The "Cantor's function result" MUST be either rational or irrational because it is a real number and those are the only possibilities for real numbers!

Organic
Hi HallsofIvy,

You wrote:
I have absolutely no idea what this means. What do you mean by "structures over scales"?
When represented by base value expansion method:

Rational numbers with infinitely many digits have repetitions over scales, for example: 0.12300123001230012300.. , or inifinitely many suffix 0 for example: 0.5000000...

Irrational numbers with infinitely many digits have NO repetitions over scales, for example:0.12360039650650110453...

I wrote:
Today math language can represent irrational numbers only by their decimal form.
My mistake. I mean today's math language can represent irrational numbers by base value expansion representation method, which is the only representation that can be used as input to Cantor's second diagonal function.

Can you show me another way that can be used as input to Cantor's second diagonal function, in his second argument ?

Thank you.

Organic

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Homework Helper
Can you show me another way that can be used as input to Cantor's second diagonal function, in his second argument ?
There are many ways to write irrational numbers. However, BECAUSE Cantor's argument uses the decimal expansion specifically, that representation is used. It's because Cantor's argument uses that expansion, not because decimal (or other place value representation) is the only way to write irrational numbers.

Actually, since real numbers can be defined as "equivalence classes of increasing sequences of rational numbers", one can do Cantor's argument much more generally: Suppose there were a list of real numbers (i.e. equivalence classes of sequences). Choose one sequence from each equivalence class. Form a new sequence as follows:
choose the nth rational number so it is NOT the nth rational number in the nth real number. One would have to show that that new sequence is not in any of the equivalence classes. I suspect that would be harder than the usual decimal form but possible.

Organic
Do you mean something like? (let us say that we don't use - and + operations)

1 <--> 1/2 3/45 7/88 ...
2 <--> 3/4 2/67 9/44 ...
3 <--> 1/9 8/17 3/37 ...
...

Then if our rule is to add 1 to the numerator then in this case Cantor's function result is: 2/2 3/67 4/37 ... which is a new sequence not in the list.

But it really doesn't matter, because still we can clime that we have some sequence as Cantor's function result, only because we deal with 2^aleph0-1 sequences, where the missing one is Cantor's function result, and this missing sequence depends on some abitrary order and the rule which we use to define Cantor's function result.

When we have 2^aleph0 sequences, Cantor's function result is unknown because we have no input at all (no rational form, no base valuse expansion form, no what so ever).

Organic

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quibton
Organic,

Very clever thinking about Cantor's irrational numbers regarding the fuzzy world of inclusion and exclusion parameters. I think you might want to include your new variable along side the current one to include the new theory you have just created. Could you try it? Think about broken symmetry theories maybe. Then compare it to one you might find along side of it. If not, then you might have created your own theory of exclusion/inclusion. Have a ago! If your theory is not anywhere else then you might have opened up a new anvenue to rationality, or not as the case may be!

Well done

quibton

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Organic
Welcome quibton,

First, thank you for your post.

I am a poor formalist, but have some ideas, which are based on structural|quantitative point of view on Math language.

They can be found here: http://www.geocities.com/complementarytheory/CATpage.html

Maybe you can help me to address these ideas in a rigorous formal way.

By doing it, we can check what idea can survive rigorous definitions.

I think that only then we can move to the next step, which is: to examine its originality.

Thank you,

Yours,

Organic

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Boolean logic is based on 0 Xor 1.

Fuzzy logic is fading between 0 Xor 1.

A non-Boolean logic is based on 0 And 1.

My point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

Number 4 is fading transition between multiplication 1*4 and addition ((((+1)+1)+1)+1) ,and vice versa.

This fading can be represented as:
Code:
(1*4)=              (1,1,1,1) <------------- Maximum symmetry-degree,
((1*2)+1*2)=        ((1,1),1,1)              Minimum information's clarity-degree (no uniqueness)
(((+1)+1)+1*2)=     (((1),1),1,1)
((1*2)+(1*2))=      ((1,1),(1,1))
(((+1)+1)+(1*2))=   (((1),1),(1,1))
(((+1)+1)+((+1)+1))=(((1),1),((1),1))
((1*3)+1)=          ((1,1,1),1)
(((1*2)+1)+1)=      (((1,1),1),1)
((((+1)+1)+1)+1)=   ((((1),1),1),1) <------ Minimum symmetry-degree,
Maximum information's clarity-degree (uniqueness)
Multiplication can be operated only between objects with structural identity .

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )

Through my point of view, there are connections between structure's symmetry-degree and information's clarity-degree.

High Entropy means maximum level of redundancy and uncertainty, which are based on the highest symmetry-degree of some system.

For example let us say that there is a piano with 3 notes and we call it 3-system :

DO=D , RE=R , MI=M

The highest Entropy level of 3-system is the most left information's-tree, where each key has no unique value of its own, and vice versa.
Code:
<-Redundancy->
M   M   M  ^<----Uncertainty
R   R   R  |    R   R
D   D   D  |    D   D   M       D   R   M
.   .   .  v    .   .   .       .   .   .
|   |   |       |   |   |       |   |   |
3 = |   |   |       |___|_  |       |___|   |
|   |   |       |       |       |       |
|___|___|_      |_______|       |_______|
|               |               |
An example of 4-notes piano:

DO=D , RE=R , MI=M , FA=F
Code:
------------>>>

F  F  F  F           F  F           F  F
M  M  M  M           M  M           M  M
R  R  R  R     R  R  R  R           R  R     R  R  R  R
D  D  D  D     D  D  D  D     D  R  D  D     D  D  D  D
.  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
|  |  |  |     |__|_ |  |     |__|  |  |     |__|_ |__|_
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|__|__|__|_    |_____|__|_    |_____|__|_    |_____|____
|              |              |              |

4 =
M  M  M
R  R                     R  R  R        R  R
D  R  D  D      D  R  D  R     D  D  D  F     D  D  M  F
.  .  .  .      .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |      |  |  |  |     |  |  |  |     |  |  |  |
|__|  |__|_     |__|  |__|     |  |  |  |     |__|_ |  |
|     |         |     |        |  |  |  |     |     |  |
|     |         |     |        |__|__|_ |     |_____|  |
|     |         |     |        |        |     |        |
|_____|____     |_____|____    |________|     |________|
|               |              |              |

D  R  M  F
.  .  .  .
|  |  |  |
|__|  |  |
|     |  |
|_____|  |
|        |
|________|
|

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