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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 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.

==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). Therefor there are as many number in <math>[0,1]</math> as <math>\mathbb{R}</math>.

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>.

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 numbers in <math>[0,1]</math> are [[uncountable]].

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

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

<math>a_{i}=1</math> when <math>a_{ii}\neq1</math> and <math>a_{i}=2</math> when <math>a_{ii}=1</math>.

<math>a_{i}=1</math> when <math>a_{ii}\neq1</math> and <math>a_{i}=2</math> when <math>a_{ii}=1</math>.

Therefore <math>a</math> is not in the list, so we have a contradition and our assumption is false, the numbers in <math>[0,1]</math> are not countable. Therefore <math>\mathbb{R}</math> is uncountable.<ref>Komolgorov, ''Introduction to Real Analysis''. (You can find it in almost any book store).</ref>

Therefore <math>a</math> is not in the list, so we have a contradiction and our assumption is false, the numbers in <math>[0,1]</math> are not countable. Therefore <math>\mathbb{R}</math> is uncountable.<ref>Komolgorov, ''Introduction to Real Analysis''. (Available in most book stores).</ref>

==Diagonalization and the Existence of God==

==Diagonalization and the Existence of God==