##  [Berry Paradox](/berry-paradox-0) 

 Definition

A self-referential paradox arising from the informal use of 'definable in under N words', typically illustrated by the phrase 'the smallest positive integer not definable in under eleven words', which yields a contradiction by naming the supposedly unnamed number.

 

 

 

 

 

 





## Principle

Principle

The organizing idea is that vagueness in natural-language definability combined with self-reference allows a single description to change the extension it purports to pick out, producing a diagonal-style contradiction between syntax and intended semantic extension.

 

 

 

 

 





## Demonstration

Demonstration

Consider the English description 'the smallest positive integer not definable in under eleven words'. If that description indeed picks out a number, it does so in fewer than eleven words, contradicting its own restriction; if it fails to pick out a number because of vagueness, the informal notion of 'definable in under eleven words' is shown to be unstable.

 

 

 

 

## Misapplication

Misapplication

Treating the paradox as evidence that natural numbers are inconsistent or that arithmetic breaks down; or applying the informal resolution to formal systems without distinguishing informal semantic definability from provability in a formal language.

 

 

 

 

 





## Consequence

Consequence

It forces a separation between informal natural-language definability and formal encodings, motivating precise syntactic encodings of 'description' and guarding against unconstrained self-referential specification in informal discourse.

 

 

 

 

## Reversal

Reversal

If one bans self-reference or explicitly quantifies over a fixed formal language in which 'definable' is interpreted, the paradox disappears; conversely, permitting only strictly stratified descriptions prevents the diagonal move that creates the contradiction.

 

 

 

 

 





## Boundary

Boundary

Applies to informal notions of 'definable' and natural-language description; it does not automatically carry over to a formal theory unless 'definability' is precisely formalized and encoded within that theory's syntax and semantics.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists with the Liar Paradox and Richard's Paradox: Berry emphasizes vagueness of 'definable' while the Liar emphasizes truth predicates; the reader must differentiate vagueness of descriptive resources from formal semantic predicates.

 

 

 

 

 





## Synthesis

Synthesis

Berry Paradox highlights that informal, imprecise claims about what can be named or defined in natural language can self-undermine when allowed to refer to the class of describable objects; resolving it requires formalizing definability or restricting self-reference.