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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 [[Real_number|real numbers]] can not be put into 1-1 correspondence to 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.

 

'''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|one-to-one correspondence]] with the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]]. This method can also be applied in other contexts, to show that two sets can't have a correspondence.  For example, it can be used to show that no set can be in 1-1 correspondence with the set of all of its subsets.

==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>.

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 this 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 this 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>

<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.<ref>Komolgorov, ''Introduction to Real Analysis''. (You can find it inalmost 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>A. N. Kolmogorov, ''Introductory Real Analysis''. ISBN 978-0486612263.</ref>

==Diagonalization and the Existence of God==

==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>

Some have cited diagonalization as a formal challenge to [[Saint Anselm]]'s [[ontological 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 [[Disprove|disproof]] of the existence of God has cast doubt on the validity of Cantor's diagonalization.

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>

==References==

==References==

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

[[Category:Mathematics]]

[[Category:Philosophy]]