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Line 1: − + − 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]]. + + − ===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>+ − 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]].+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − <references/>+ − +
Fixed this load of absolute 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 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.
==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>.
We will now use [[proof by contradiction]] to show that the numbers in <math>[0,1]</math> are uncountable.
Assume that that they are countable then we can list them as such,
<math>
0.a_{11}a_{12}a_{13}a_{14}a_{15}\dots
</math>
<math>
0.a_{21}a_{22}a_{23}a_{24}a_{25}\dots
</math>
<math>
0.a_{31}a_{32}a_{33}a_{34}a_{35}\dots
</math>
<math>
0.a_{41}a_{42}a_{43}a_{44}a_{45}\dots
</math>
<math>
\vdots
</math>
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 ==
Komolgorov, ''Introduction to Real Analysis''. (You can find it inalmost any book store).
[[Category:Mathematics]][[Category:Philosophy]]
[[Category:Mathematics]]