Diagonalization - Revision history

DavidB4-bot: Spelling/Grammar Check, typos fixed: Therefore → Therefore, (3)

2016-07-15T15:48:19Z

Spelling/Grammar Check, typos fixed: Therefore → Therefore, (3)

PatrickD: Put back corrected version of sentence that I had removed.

2009-10-22T21:40:23Z

Put back corrected version of sentence that I had removed.

PatrickD: Remove apparent parody dating to very first version. There is no "disproof of the existence of God" here.

2009-10-22T02:50:12Z

Remove apparent parody dating to very first version. There is no "disproof of the existence of God" here.

PatrickD: Rewrite intro; make the mapping explicit, so we don't link into pages that aren't helping the cause.

2009-10-22T01:47:18Z

Rewrite intro; make the mapping explicit, so we don't link into pages that aren't helping the cause.

PatrickD: Give ISBN number rather than (non-encyclopedic) "available in most bookstores", spell author correctly, spell title correctly.

2009-10-20T17:17:24Z

Give ISBN number rather than (non-encyclopedic) "available in most bookstores", spell author correctly, spell title correctly.

Foxtrot: fixed link

2008-06-21T23:52:35Z

fixed link

Foxtrot: links, typos

2008-06-21T23:51:11Z

links, typos

CSGuy at 23:46, June 20, 2008

2008-06-20T23:46:22Z

CSGuy at 23:45, June 20, 2008

2008-06-20T23:45:38Z

DanielB at 23:38, June 20, 2008

2008-06-20T23:38:26Z

@@ Line 45: / Line 45: @@
 <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>
+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>
+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>

@@ Line 1: / Line 1: @@
 {{Math-h}}
-'''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|1-1 correspondence]] with the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]].  This method can also be applied in other contexts.
+'''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==
@@ Line 12: / Line 12: @@
 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 open interval has no such correspondence, and thus it, and the real line as a whole, 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 the open interval are in a 1-1 correspondence with the natural numbers.  Then we can make an (infinite) sequential list of them, like this:
+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>

@@ Line 51: / Line 51: @@
 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 [[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==
@@ Line 57: / Line 57: @@
 {{reflist|2}}
-[[Category:Mathematics]][[Category:Philosophy]]
+[[Category:Mathematics]]

@@ Line 1: / Line 1: @@
-'''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 1-1 correspondence to the [[natural number]]s, thereby demonstrating the real numbers are not [[countable]]. This method can be applied to any infinite set to construct an even larger infinite set.
+{{Math-h}}
 ==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). Therefore 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:
-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 the 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 the 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>

@@ Line 37: / Line 37: @@
 <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>Komolgorov, ''Introduction to Real Analysis''. (Available in most book stores).</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==

← Older revision		Revision as of 23:46, June 20, 2008
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 [[~~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 1-1 correspondence to the [[natural number]]s, thereby demonstrating the real numbers are not countable. This method can be applied to any infinite set to construct an even larger infinite set.

	==Proof of the non-countability of real numbers==		==Proof of the non-countability of real numbers==

← Older revision		Revision as of 23:38, June 20, 2008
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 [[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\|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.

	==Proof of the non-countability of real numbers==		==Proof of the non-countability of real numbers==