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reinstating old material. Cited and not rubbish.
'''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.
'''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==
==Proof of the non-countability of real numbers==
<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>.
Therefor <math>a</math> is not in the list, so we have a contradition and our assumption is false, the number <math>[0,1]</math> are not countable. Therefor <math>\mathbb{R}</math> is uncountable.
Therefor <math>a</math> is not in the list, so we have a contradition and our assumption is false, the number <math>[0,1]</math> are not countable. Therefor <math>\mathbb{R}</math> is uncountable.<ref>Komolgorov, ''Introduction to Real Analysis''. (You can find it inalmost any book store).</ref>
== References ==
==Diagonalization and the Existence of God==
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>
[[Category:Mathematics]]
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 [[Disprove|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]].
==References==
{{reflist|2}}
[[Category:Mathematics]][[Category:Philosophy]]