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

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

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

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

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

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

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

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

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.