Tarski's undefinability theorem
E71179
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Tarski’s undefinability theorem | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in mathematical logic ⓘ |
| appliesTo |
Peano arithmetic
ⓘ
arithmetically adequate theories ⓘ first-order arithmetic ⓘ formal languages ⓘ sufficiently strong formal systems ⓘ |
| concerns |
definability of semantic concepts
ⓘ
limitations of formal systems ⓘ truth predicates ⓘ |
| consequence |
no formula in the language of arithmetic can capture exactly the Gödel numbers of all true arithmetic sentences
ⓘ
truth in arithmetic is not arithmetically definable ⓘ truth is strictly stronger than provability in arithmetic ⓘ |
| field |
mathematical logic
ⓘ
metalogic ⓘ model theory ⓘ |
| formalizes | impossibility of defining a global truth predicate for a language within itself ⓘ |
| historicalPeriod | 20th-century logic ⓘ |
| holdsIn | any consistent, sufficiently strong, effectively axiomatizable theory extending Robinson arithmetic ⓘ |
| implies |
hierarchy between object language and metalanguage
ⓘ
nonexistence of an internal truth predicate for arithmetic ⓘ semantic notions like truth may require a stronger metalanguage ⓘ |
| influenced |
philosophy of language
ⓘ
philosophy of mathematics ⓘ theories of truth in analytic philosophy ⓘ |
| mainTopic |
formal theories of truth
ⓘ
undefinability of truth ⓘ |
| motivated |
development of formal truth theories
ⓘ
distinction between object language and metalanguage in logic ⓘ study of truth hierarchies ⓘ |
| namedAfter | Alfred Tarski ⓘ |
| relatedTo |
Gödel's incompleteness theorems
ⓘ
Tarskian object-language/metalanguage distinction ⓘ
surface form:
Tarski's hierarchy of languages
Tarskian object-language/metalanguage distinction ⓘ
surface form:
Tarski's semantic conception of truth
Tarski–Mostowski–Robinson theorem ⓘ definability theory ⓘ liar paradox ⓘ |
| requires |
effective axiomatizability of the theory
ⓘ
sufficient expressive power to represent arithmetic ⓘ |
| statesThat |
no arithmetically definable predicate in the language of arithmetic can coincide with the truth predicate for arithmetic
ⓘ
there is no formula in the language of arithmetic that defines the set of all true arithmetic sentences ⓘ truth for the language of a sufficiently strong theory cannot be defined within that same language ⓘ |
| typeOf |
metatheorem about formal theories
ⓘ
undefinability result ⓘ |
| uses |
arithmetization of syntax
ⓘ
diagonalization ⓘ self-referential sentences ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Tarski’s undefinability theorem