Difference between revisions of "Diagonalization"

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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 the real numbers are not countable.  
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{{Math-h}}
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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 [[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==
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There exists a map <math>f:[0,1]\rightarrow\mathbb{R}</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>.
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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:
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:<math>y = \frac{\tan^{-1}(x)}{\pi} + \frac{1}{2}</math>
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maps the entire real line to the open interval <math>(0, 1)\,</math>.  Its inverse:
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:<math>x = \tan(\pi(y - 1/2))\,</math>
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maps the open interval to the entire real line.
  
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We will now use [[proof by contradiction]] to show that the numbers in <math>[0,1]</math> are uncountable.
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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.
  
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Assume that that they are countable then we can list them as such,
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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]].
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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>
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<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>.
  
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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.
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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>
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==Diagonalization and the Existence of God==
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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>
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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>
  
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== References ==
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==References==
  
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Komolgorov, ''Introduction to Real Analysis''. (You can find it inalmost any book store).
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{{reflist|2}}
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]
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[[Category:Philosophy]]

Latest revision as of 15:48, July 15, 2016

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Diagonalization is a technique first used by Georg Cantor, a German mathematician. He used it to show that the real numbers can not be put into one-to-one correspondence with the natural numbers, 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

First, we create a 1-1 correspondence between the entire real line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}\,} and the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1)\,} . This function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{\tan^{-1}(x)}{\pi} + \frac{1}{2}}

maps the entire real line to the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1)\,} . Its inverse:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \tan(\pi(y - 1/2))\,}

maps the open interval to the entire real line.

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0, 1)\,} is in correspondence.

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 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.a_{11}a_{12}a_{13}a_{14}a_{15}\dots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.a_{21}a_{22}a_{23}a_{24}a_{25}\dots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.a_{31}a_{32}a_{33}a_{34}a_{35}\dots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.a_{41}a_{42}a_{43}a_{44}a_{45}\dots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vdots }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}\in\{0,1,2,3,4,5,6,7,8,9\}}

Construct the number,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=0.a_{1}a_{2}a_{3}a_{4}\dots} , where


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{i}=1} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ii}\neq1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{i}=2} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ii}=1} .

Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is not in the list, so we have a contradiction and our assumption is false, the numbers in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1]} are not countable. Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} is uncountable.[1]

Diagonalization and the Existence of God

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.[2]

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.[3]

References

  1. ↑ A. N. Kolmogorov, Introductory Real Analysis. ISBN 978-0486612263.
  2. ↑ http://www.ephilosopher.com/e107_plugins/forum/forum_viewtopic.php?104130
  3. ↑ 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.
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