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## Main Question or Discussion Point

Hi,

Please read it until the end, before you write your remarks.

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 . 0

0 . 4 2

0 . 1 0 1

0 . 3 3 3 3

2 . 0 0 0 0 0

0 . 3 5 4 9 5 5

3 . 0 0 0 0 0 0 0

0 . 6 4 1 6 4 1 6 4

0 . 3 0 2 0 3 0 2 0 3

0 . 6 1 3 6 1 3 6 1 3 6

0 . 2 7 1 0 2 7 1 0 2 7 1

...

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

Please read it until the end, before you write your remarks.

Thank you.

----------------------------------------------------------------------------

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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