Topic: Godel's incompleteness theorem
Topic: infinity and infinitesimal
Topic: limitations of formalism
Topic: self reference
Topic: semantic truth; s iff p
Topic: what is truth
 
Summary
Consider the proposition (1) (1) is not true. If true, the statement is false by definition; if false, the statement is true. This selfreferential statement is an instance of the liar's paradox. It's very existence denies its truth.
Russell discovered the first Liar's paradox, the set of all predicates that are not true of themselves. Does this set belong to itself? If true, it is false; if false, true. Here is a definable collection that can not be defined. Godel used the Liar's paradox to prove that a complete system of logic contains true statements that are definable in the system, but not proveable.
Both logic and computers produce infinite consequences from finite definitions. These paradoxes are like an infinite loop or infinite regress  If true, false; if false, true; if true,false; ... Do these paradoxes really limit the usefulness of logic and computers? (cbb 6/06)
Subtopic: Russell's paradox
Quote: let w be the predicate: to be a predicate that cannot be predicated of itself; can w be predicated of itself [»russB6_1902]
 Quote: there is no class (as a totality) of those classes that do not belong to themselves
 Quote: a definable collection does not necessarily form a totality
 Quote: abstraction leads to Russell's paradox, (.thereExists.x)((x.in.x).equiv. ~(x.in.x))
 Quote: Russell's paradox shook the foundations of Frege's arithmetic; substitution of equality is not always valid [»russB6_1902]
 Quote: Russell's theory of types avoids Russell's paradox by stratified formulas; e.g., .in. only occurs in contexts of the form n .in. n+1
 Quote: avoid Russell's paradox by restricting abstraction to stratified conditions that do not reference the abstract class [»quinWV2_1937]
 Quote: avoid Russell's paradox by separate rules for class existence and elementhood
 Subtopic: liar's paradox
Note: the Liar's paradox: (1) (1) is not true [»tarsA_1944, OK]
 Quote: normally a contradiction is a criterion for having done something wrong; not for the Liar's paradox
 Quote: the antinomies of formal logic arise because classifications must be modified for unforeseen objects [»poinH_1908, OK]
 Subtopic: liar's paradox as a game
Quote: the Liar's paradox is just a language game that behaves differently than others [»wittL_1939]
 Subtopic: truth and the liar's paradox
Quote: any treatment of the concept of truth must circumvent the liar's paradox, e.g., the Cretan prophet [»kripS_1975]
 Quote: many ordinary assertions about truth are liable to the liar's paradox if empirical facts are extremely unfavorable [»kripS_1975]
 Subtopic: examples of the liar's paradox
Quote: Godel's sentence is similar to the Liar's paradox and Richard's antimony
 Quote: the Godel sentence G, if it were a true theorem, would state "G is not a theorem" [»hofsDR_1979]
 Quote: Godel's string is equivalent to "I can not be proved in the formal system" [»hofsDR_1979, OK]
 Quote: the following sentence is false; the preceding sentence is true [»hofsDR_1979]
 Quote: let 'Jack' be a name of the sentence 'Jack is short'; could use to prove Godel's incompleteness theorem [»kripS_1975]
 Quote: let w be the predicate: to be a predicate that cannot be predicated of itself; can w be predicated of itself [»russB6_1902]
 Subtopic: avoiding liar's paradox
Quote: prevent Liar's paradox by not using a semantically closed language with 'true' and "X is true iff p"; use metalanguage and object language [»tarsA_1944]
 Quote: a sentence is grounded if one can determine its truth from sentences not mentioning truth; liar's paradox is not grounded [»kripS_1975]
 Quote: grounded sentences define a fixed point, a language that contains its own truth predicate; liar paradoxes are ungrounded [»kripS_1975]
 Quote: Tarski's hierarchy of languages is the orthodox approach to the liar's paradox; but it requires many versions of 'true' [»kripS_1975]
 Quote: state regularities do not occur for nonsense sentences, nor for sentences used in the liar paradox [»ziffP_1960]

Related Topics
Topic: Godel's incompleteness theorem (19 items)
Topic: infinity and infinitesimal (37 items)
Topic: limitations of formalism (93 items)
Topic: self reference (27 items)
Topic: semantic truth; s iff p (34 items)
Topic: what is truth (67 items)
