Difference between revisions of "Diagonalization"

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

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This article/section deals with mathematical concepts appropriate for late high school or early college.

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 $(0,1)\,$ . This function:

y={\frac {\tan ^{-1}(x)}{\pi }}+{\frac {1}{2}}

maps the entire real line to the open interval $(0,1)\,$ . Its inverse:

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 $(0,1)\,$ 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:

$0.a_{11}a_{12}a_{13}a_{14}a_{15}\dots$

$0.a_{21}a_{22}a_{23}a_{24}a_{25}\dots$

$0.a_{31}a_{32}a_{33}a_{34}a_{35}\dots$

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 }

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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

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

[1] A. N. Kolmogorov, Introductory Real Analysis. ISBN 978-0486612263.

[2] ttp://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.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-'''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.
+{{Math-h}}
-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''' 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.
-===Diagonalization and the Existence of God===
+==Proof of the non-countability of real numbers==
+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>
+maps the open interval to the entire real line.
-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>
+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.
-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]].
+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]].
-== References ==
+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:
-<references/>
+<math>
+.a_{11}a_{12}a_{13}a_{14}a_{15}\dots
+</math>
-[[Category:Mathematics]][[Category:Philosophy]]
+<math>
+.a_{21}a_{22}a_{23}a_{24}a_{25}\dots
+</math>
+<math>
+.a_{31}a_{32}a_{33}a_{34}a_{35}\dots
+</math>
+<math>
+.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>.
+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==
+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>
+==References==
+{{reflist|2}}
+[[Category:Mathematics]]
+[[Category:Philosophy]]