Undecidable - Revision history

VargasMilan: /* Undecidability and Faith */ + umlaut

2018-03-24T08:29:15Z

Undecidability and Faith: + umlaut

VargasMilan: typo

2018-03-24T08:25:28Z

typo

DavidB4-bot: /* Famous Undecidable Statements */clean up & uniformity

2016-07-13T21:01:02Z

Famous Undecidable Statements: clean up & uniformity

DavidB4-bot: /* top */Spelling/Grammar Check, typos fixed: etc) → etc.)

2016-06-27T18:44:36Z

top: Spelling/Grammar Check, typos fixed: etc) → etc.)

RolandPlankton: link to cardinal number rather than to cardinal; latter is a high-ranking Roman catholic priest

2012-03-21T23:04:10Z

link to cardinal number rather than to cardinal; latter is a high-ranking Roman catholic priest

Aroth at 01:49, September 15, 2009

2009-09-15T01:49:38Z

MarkGall at 01:02, September 15, 2009

2009-09-15T01:02:58Z

Aroth: Added section on undecidability and faith, as discussed on the talk page.

2009-09-15T00:50:03Z

Added section on undecidability and faith, as discussed on the talk page.

Conservative: Reverted edits by BushBud (Talk) to last version by CSGuy

2009-05-14T00:55:02Z

Reverted edits by BushBud (Talk) to last version by CSGuy

BushBud: Undecidability justifies faith

2009-05-14T00:51:56Z

Undecidability justifies faith

@@ Line 4: / Line 4: @@
 ==Undecidability and Faith==
-The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.
+The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Gödel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.
 ==Famous Undecidable Statements==

← Older revision		Revision as of 08:25, March 24, 2018
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−	A statement in formal [[logic]] is called '''undecidable''' if there is no [[proof]] or [[disproof]] of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be ~~undeciable~~ in [[Zermelo-Fraenkel]] set theory) is actually false.	+	A statement in formal [[logic]] is called '''undecidable''' if there is no [[proof]] or [[disproof]] of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be undecidable in [[Zermelo-Fraenkel]] set theory) is actually false.

	As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc.), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.		As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc.), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.

← Older revision		Revision as of 21:01, July 13, 2016
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	* [[Russell's paradox]]		* [[Russell's paradox]]

−	[[~~category~~:~~logic~~]]	+	[[Category:Logic]]

@@ Line 1: / Line 1: @@
 A statement in formal [[logic]] is called '''undecidable''' if there is no [[proof]] or [[disproof]] of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be undeciable in [[Zermelo-Fraenkel]] set theory) is actually false.
-As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.
+As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc.), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.
 ==Undecidability and Faith==

@@ Line 11: / Line 11: @@
 * [[Banach-Tarski paradox]]
 * [[Continuum hypothesis]]
-* Existence of large [[cardinal]]s
+* Existence of large [[cardinal number]]s
 * [[Halting problem]]
 * [[König's lemma]]

@@ Line 19: / Line 19: @@
 * [[Ramsey theory]] involving infinite sets
 * [[Russell's paradox]]
 [[category:logic]]

← Older revision		Revision as of 00:50, September 15, 2009
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	As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.		As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible [[atom]]ic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.
		+
		+	==Undecidability and Faith==
		+	The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.

	==Famous Undecidable Statements==		==Famous Undecidable Statements==

← Older revision		Revision as of 00:55, May 14, 2009
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	[[category:logic]]		[[category:logic]]
−
−
−	~~==Undecidability and God==~~
−	The theory of undecidability shows that the world cannot be guided by [[logic]] and [[rationality]] alone: Godel proved that there exist truths that can never be proven. This demonstrates the validity of [[faith]].

← Older revision		Revision as of 00:51, May 14, 2009
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	[[category:logic]]		[[category:logic]]
		+
		+
		+	==Undecidability and God==
		+	The theory of undecidability shows that the world cannot be guided by [[logic]] and [[rationality]] alone: Godel proved that there exist truths that can never be proven. This demonstrates the validity of [[faith]].