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'''Diagonalization''' is a technique first used by [[Cantor|George Cantor]], a [[Germany|German]] [[Mathematician|mathematician]]. He used it to show that the [[Cardinality|cardinality]] of the [[Real_number|real numbers]] is not equal to the cardinality of the [[Natural_number|natural numbers]], thereby demonstrating one [[Infinity|infinity]] is bigger than the other infinity.  

'''Diagonalization''' is a technique first used by [[Cantor|George Cantor]], a [[Germany|German]] [[Mathematician|mathematician]]. He used it to show that the [[Cardinality|cardinality]] of the [[Real_number|real numbers]] is not equal to the cardinality of the [[Natural_number|natural numbers]], thereby demonstrating the real numbers are not countable.  

This method can be applied to any infinite set to construct an even larger infinite set. The method uses the [[Axiom_of_Choice|Axiom of Choice]].  

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

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

===Diagonalization and the Existence of God===

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

Some have cited diagonalization as a formal challenge to [[Saint_Anselm|Saint Anselm]]'s argument for the existence of God. In summary, Anselm argued that there must be a greatest idea and what could be greater than God? Therefore God exists.<ref>http://www.ephilosopher.com/e107_plugins/forum/forum_viewtopic.php?104130</ref>

Assume that that they are countable then we can list them as such,

However, diagonalization argues that no greatest idea can exist: quite bluntly, God is infinite, therefore He can be diagonalized to produce an even greater infinite.<ref>''Topo-philosophies: Plato's Diagonals, Hegel's Spirals, and Irigaray's Multifolds'', Arkady Plotnitsky. In ''After Poststructuralism: Writing the Intellectual History of Theory'' Tilottama Rajan, Michael James.</ref> This seeming [[disproof]] of the existence of God has cast doubt on the validity of Cantor's diagonalization. In particular, it provides yet another counterintuitive conclusion from the dubious [[Axiom_of_Choice|Axiom of Choice]].

 

 

 

 

 

Where <math>a_{ij}\in\{0,1,2,3,4,5,6,7,8,9\}</math>

 

Construct the number,

 

<math>a=0.a_{1}a_{2}a_{3}a_{4}\dots</math>, where

 

 

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

 

Therefor <math>a</math> is not in the list and so we have a contradition so our assumption is false hand the number <math>[0,1]</math> are not countable. Therefor <math>\mathbb{R}</math> is uncountable.

== References ==

== References ==

[[Category:Mathematics]][[Category:Philosophy]]

[[Category:Mathematics]]