# A question on Cantor's second diagonalization argument

Hurkyl
Staff Emeritus
Gold Member
.010101010... (1/99) is not in the list exactly as Cantor's function real number result is not in the list.
Then why are you claiming:

My list is a decimal representation of any rational number in Cantor's first argument specific list.
When there is clearly at least one rational number whose decimal representation is not in your list?

This is the exact state between Cnator's function some result and the R list.
But there's a very important logical piece you're missing. Cantor's second diagonal argument is applied to EVERY list of real numbers, and that's how you conclude |N|<|R|.

However, you are applying the diagonal argument to a single list, and if it is the case that no rational number appears twice on the list, all we can conclude is |N|<=|Q|.

That is the key you are missing. In order to prove |S|<|T| for sets S and T, you have to prove that EVERY FUNCTION from S to T is missing an element of t. Proving it for only one function doesn't cut it1.

1: except when there exists only one function from S to T, in which case proving it for one function coincides with proving it for every function. This situation can only happen when S or T is the empty set, or when |S|=|T|=1.

Therefore there are no limits to the number of ways that Q list can be rearranged to give us some new Q number (which is not in the list) as a Cantor's function result.
Sure there is. There are at most |N|^|N| = c ways.

And I bet I can give you a list of rational numbers for which "Cantor's function" does not give you a rational number for any rearrangement.

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Hi Hyrkyl,

Sure there is. There are at most |N|^|N| = c ways.

And I bet I can give you a list of rational numbers for which "Cantor's function" does not give you a rational number for any rearrangement.
Ok, I'm waiting.

Thank you Hurkyl, my argument is wrong because I have at least one rational decimal form which is not in Cantor's diagonal first list.

Therefore, we can conclude that at least one rearrangement of all rational numbers decimal forms, can't give us a rational number as a result.

This is a beautiful insight for me, and you are a great mentor.

Organic

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HallsofIvy
Homework Helper
Can Hurkyl be as vague in defining his list as you are in yours?

Thank you Hurkyl, my argument is wrong because I have at least one rational decimal form which is not in my list, but exists in Cantor's diagonal first list.

Therefore, we can conclude that at least one rearrangement of all rational numbers decimal forms, can't give us a rational number as a result.

This is a beautiful insight for me, and you are a great mentor.

Organic

So, I have a question.

Cantor's first diagonal argument is like the Rosetta stone (http://www.rosetta.com/RosettaStone.html [Broken]) for the decimal forms of rational numbers.

Do Irrational numbers can have some kind of "Rosetta Stone" ?

Organic

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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 state, 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 ?

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

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

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 talking about the structural difference between |R| and |R|-1.

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HallsofIvy
Homework Helper
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.
No, we call R "uncountable" because it fits the precise mathematical definition of the word "uncountable".

I know that 2^aleph0-1 = 2^aleph0, but again I talking about the structural difference between |R| and |R|-1.
Then perhaps it would be a good idea for you to tell us exactly what YOU mean by "|R|-1", as well as what you mean by the "structure" of a cardinality.

Hi HallsofIvy,

You wrote:
...it would be a good idea for you to tell us...
How is us ? Are you more then a one person ?

And if not, it looks like you think that you are the speaker of some group.

In this case what is the name of the group that gave you the job to be their speaker ?

My advice to you is: Please be more modest and less aggressive to the persons who write their point of views on any subject in this forum. You can learn it from Hurkyl.

First of all Math is a form of language and any language is a communication tool between persons, even if its gremial is rigorous. Please don't forget this important point.

Some persons are professionals and some are not, but you never know where some good idea can appear.

If you understand what I wrote about the structural difference between aleph0 and aleph0-1, then take this understanding and connect it to 2^aleph0 and 2^aleph0-1.

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 R number (repetitions, or no repetitions over scales).

When the list has 2^aleph0 numbers, then Cantor's function result is unknown.

Organic

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HallsofIvy
Homework Helper
Look up the phrase "editorial we". Yes, I was speaking on behalf of all the people (one or two at least) who read this thread. I certainly don't expect you to correspond with me personally to define your terms. I have no doubt that anyone reading this thread would appreciate you actually defining your terms.

You are completely correct that Hurkyl is "more modest and less aggressive" than I am. He perhaps still has some hopes that you will actually understand what he is saying.

I might also point out that you are not a good one to talk about being "more modest and less aggressive". You have repeatedly asserted that you knew for a fact that something that all professional mathematicians understand and accept is incorrect. You then give a lot of undefined terms and hand-waving to support that contention. Your basic argument appears to be "I do not understand this, therefore it is wrong."

HallsofIvy,

Pay attention to the fact that you did not write a single word of what I wrote, after what I wrote personally to you.

Because you are a professional mathematician, which is the mentor of the Homework help zone, I think that you can do more then just repeating on the words "non-sense".

For example, Hurkyl helped me to address some idea of mine in a formal definition.