Berry paradox
E73102
The Berry paradox is a self-referential logical paradox arising from phrases like “the smallest positive integer not definable in under eleven words,” which appears to define exactly such a number while claiming it cannot be defined.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Berry’s paradox | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
logical paradox
ⓘ
philosophical paradox ⓘ self-referential paradox ⓘ semantic paradox ⓘ |
| category |
paradoxes of definability
ⓘ
paradoxes of self-reference ⓘ |
| concerns |
finite descriptions of numbers
ⓘ
positive integers ⓘ |
| describedBy | the phrase "the smallest positive integer not definable in under eleven words" ⓘ |
| field |
foundations of mathematics
ⓘ
logic ⓘ mathematical logic ⓘ philosophy of mathematics ⓘ |
| hasAlternativeName |
Berry paradox
ⓘ
surface form:
Berry’s paradox
|
| hasKeyFeature |
arises from quantifying over all definitions expressible in a language
ⓘ
can be avoided by formalizing the notion of definition ⓘ depends on informal notions of definition and word length ⓘ illustrates limitations of naive talk about definability ⓘ motivates precise meta-mathematical frameworks ⓘ shows tension between arithmetic and natural language descriptions ⓘ uses a phrase that appears to define a number while asserting it is not definable ⓘ |
| historicalNote |
based on an observation attributed to G. G. Berry
ⓘ
discussed by Bertrand Russell ⓘ |
| illustrates |
the need to distinguish object language from meta-language
ⓘ
the non-absolute nature of definability ⓘ |
| influenced |
development of algorithmic information theory
ⓘ
studies of definability in arithmetic ⓘ |
| involvesConcept |
arithmetization of language
ⓘ
definability ⓘ description length ⓘ liar-type construction ⓘ meta-language ⓘ natural language ⓘ self-reference ⓘ |
| namedAfter | G. G. Berry ⓘ |
| relatedTo |
Kolmogorov complexity
ⓘ
surface form:
Chaitin’s incompleteness theorem
Grelling–Nelson paradox ⓘ Gödel's incompleteness theorems ⓘ
surface form:
Gödel’s incompleteness theorems
Kolmogorov complexity ⓘ Richard paradox ⓘ Russell’s paradox ⓘ definability in arithmetic ⓘ liar paradox ⓘ semantic paradoxes ⓘ set-theoretic definability ⓘ |
| resolutionApproach |
formalization of the notion of definition
ⓘ
use of precise syntactic measures instead of informal word counts ⓘ |
| usedInArgument |
arguments about the limits of formal systems
ⓘ
arguments for the necessity of a hierarchy of languages ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Berry’s paradox