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'''Diagonalization''' is a technique first used by [[Cantor|Georg Cantor]], a [[Germany|German]] [[mathematician]]. He used it to show that the [[real number]]s can not be put into 1-1 correspondence to the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]]. This method can be applied to any infinite set to construct an even larger infinite set.

 

'''Diagonalization''' is a technique first used by [[Cantor|Georg Cantor]], a [[Germany|German]] [[mathematician]]. He used it to show that the [[real number]]s can not be put into [[bijection|1-1 correspondence]] with the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]]. This method can also be applied in other contexts.

==Proof of the non-countability of real numbers==

==Proof of the non-countability of real numbers==

There exists a map <math>f:\mathbb{R}\rightarrow[0,1]</math> (in fact all infinitly supported [[probability distribution]] does this). Therefore there are as many number in <math>[0,1]</math> as <math>\mathbb{R}</math>.

First, we create a 1-1 correspondence between the entire real line <math>\mathbb{R}\,</math> and the open interval <math>(0, 1)\,</math>.  This function:

:<math>y = \frac{\tan^{-1}(x)}{\pi} + \frac{1}{2}</math>

maps the entire real line to the open interval <math>(0, 1)\,</math>.  Its inverse:

:<math>x = \tan(\pi(y - 1/2))\,</math>

 

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval <math>(0, 1)\,</math> is in correspondence.

We will now use [[proof by contradiction]] to show that the numbers in <math>[0,1]</math> are [[uncountable]].

We will now use [[proof by contradiction]] to show that the open interval has no such correspondence, and thus it, and the real line as a whole, are [[uncountable]].

Assume the numbers in [0,1], are countable. Then we can list them as such,

Assume the numbers in the open interval are in a 1-1 correspondence with the natural numbers. Then we can make an (infinite) sequential list of them, like this:

<math>

<math>