Countability of the real numbers


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Published: 2016-04-04


Countability of the real numbers

aka. Disproving Cantor Diagonal Argument

 

Cantor Diagonal Argument:

  1. A set is countable if, and only if, we can associate to the set, a function that is a bijection from each element of the set, to natural numbers

  2. If the real numbers ("R") were countable, then a subset would be, like the interval ]0-1[

  3. Cantor argument on a finite set:

    1. Let A be a finite set. Then there is no bijection between A and the set of sequences (s_a), a in A, where s_a is 0 or 1

    2. Because if the first set has "n" elements, then the second set would have "2^n" elements

    3. And therefore, the sets being of different sizes, they can't have a bijection

    4. (no diagonal argument needed)

    5. (and then what? Is {1,2,3} uncountable?)

  4. Cantor argument on the interval ]0-1[:

    1. Let A be the infinite set of real numbers between 0 and 1. Then there is no bijection between A and the infinite set of sequences (s_a), a in A, where s_a is 0 or 1

    2. Because if the first set has "infinitely many" elements, then the second set would have "2^(infinitely many)" elements

    3. And therefore, the sets being of different sizes, they can't have a bijection

    4. (no diagonal argument needed)

    5. (and then what? Is ]0-1[ uncountable?)

 

 

Counting real numbers of the interval ]0-1[:

  1. A set is countable if, and only if, we can associate to the set, a function that is a bijection from each element of the set, to natural numbers

  2. Let's imagine a function C1 that:

    1. To real numbers 0.1 to 0.9, associates numbers 11 to 19,

    2. To real numbers 0.01 to 0.09, associates numbers 101 to 109,

    3. To real numbers 0.11 to 0.19, associates numbers 111 to 119,

    4. ...

    5. To real numbers 0.91 to 0.99, associates numbers 191 to 199,

    6. ...

    7. To real numbers 0.001 to 0.009, associates numbers 1001 to 1009,

    8. ...

  3. What the function C1 is doing, in the writing of the real numbers in base 10, is to replace the part "0." with a "1".

    1. So, that function successfully associates to each real number, a unique natural number,

    2. Fractional numbers have a unique representation in this function,

    3. Irrational numbers also have a unique representation,

      1. For example: sqr(2)/2 = 0.707107... => 1707107...

  4. Let's imagine a function C2 that:

    1. To real numbers 0.1 to 0.9, associates numbers 1 to 9,

    2. To real numbers 0.01 to 0.09, associates numbers 10 to 90 (adding 10 at each step),

    3. To real numbers 0.11 to 0.19, associates numbers 11 to 91 (adding 10 at each step),

    4. ...

    5. To real numbers 0.91 to 0.99, associates numbers 19 to 99 (adding 10 at each step),

    6. ...

    7. To real numbers 0.001 to 0.009, associates numbers 100 to 900 (adding 100 at each step),

    8. ...

  5. What the function C2 is doing, in the writing of the real numbers in base 10, is to remove the part "0.", and reverse the following digits.

    1. So, that function successfully associates to each real number, a unique natural number,

    2. Fractional numbers have a unique representation in this function,

    3. Irrational numbers also have a unique representation,

      1. For example: sqr(2)/2 = 0.707107... => ...701707

  6. So we have successfully associated a counting function, to the set of real numbers of that interval ]0-1[

  7. Real numbers are countable

  8. Cantor "uncountable infinity" "bigger than infinity" does not exist

 

 

Remarks:

  1. Answering the remark "But your functions C1 and C2 are dealing with infinitely big numbers, written with infinitely many digits. You can't do that". Oh, really, can't I?

    1. Think about this numbers: 0.3333... Is it well defined in this notation? Yes it is. And how many digits do this number have? Infinitely many. So, no problem with infinity, it's a real number

    2. Then why shouldn't I use the C1 transformation and write: 3333... ? Well, this is not allowed because the number is infinite to the left, so it should more be written as: ...3333 but this notation is not already used in Maths, even if it makes perfect sense.

    3. So the solution is to write the number as the infinitely many digits he is, regardless of a notation (in a sum "to the right"):

      1. Sumn[1-∞](3/10^n)

    4. And then transform the digits in C1 (in a sum "to the left"):

      1. Sumn[1-∞](3.10^n)

  2. And if for a reason I don't get, the last notation is disallowed, I produced the C2 function which is putting the infinitely many digits "to the right" again

 

 

Cantor's error analysis:

  1. Cantor's error is to suppose, without saying it, that the "infinitely long" list of the digits and the "infinitely long" list of the real numbers have the same length, "infinity"

  2. Then from that point, using a diagonal trick, he can imagine a new entity out of the list, which proves, he is right on that, that the initial unsaid supposition was wrong, and that the number of digits is not the same than the number of numbers. But of course it isn't!

  3. The number of digits is not the same than the number of numbers, because for each digit you write, you have 10 times more numbers!

  4. So Cantor is not proving that uncountable infinity does exist, he is just proving that he can assume something wrong without saying it, then disprove himself, then screw humanity with his own mental illness of seeing "infinity" as a number.

  5. Infinity is not a thing, it's a dimension, in which things are defined.

  6. Nothing contains a dimension. It's a dimension, that is containing things.

 

 

Remarks:

  1. Cantor, with his diagonal trick, can find only one number out of the list he is considering.

  2. I can find infinitely many more! Because when "n" digits are in the list, the list is "10^n" long, so that the list misses "(10^n)-n" numbers.

  3. So with "n" taken to infinity, Cantor's list misses about 10^infinitely many numbers!