Berry's Paradox
"The least integer not nameable in fewer than nineteen syllables" — eighteen syllables
First published: B. Russell, "Les paradoxes de la logique", *Revue de Métaphysique et de Morale* 14 (1906): 627–650.
Consider "the smallest positive integer not nameable in fewer than nineteen syllables." That phrase has eighteen.
Berry's paradox: the phrase "the smallest positive integer not nameable in fewer than nineteen syllables" has only eighteen syllables. So the integer it names is nameable in fewer than nineteen syllables — contradicting its definition. The paradox is semantic rather than set-theoretic (no quantification over sets), and structurally identical to Gödel and Tarski: self-referential definability cannot be unrestricted. Standard resolutions invoke Tarskian hierarchies and restrict "nameable" to a specific formal language.
Formulation
Let n = "the smallest positive integer not nameable in fewer than nineteen syllables." The defining phrase has 18 syllables. If n exists, it is nameable in 18 syllables — contradicting the definition. Conclusion: unrestricted use of "nameable" generates contradiction.
Dimensions Engaged
Information
Engages Information · Granularity: self-referential definability produces paradox unless restricted.
Responses — How Schools Engage
Affirms / takes the bait 4
A canonical semantic paradox; alongside Liar, Curry, Grelling, it forces refinement of truth and definability across formal languages.
Operational regimentation: "nameable" must be relativised to a specific language; unrestricted use violates the verification standards positivism requires.
A clean structural argument for hierarchies in semantics: each language's "nameable" predicate cannot be defined within that language.
Impredicative definitions generate paradox; constructive caution about self-referential definability is empirically vindicated.
Reframes the question 2
Numbers and their properties are independent of how we name them; Berry's paradox is about naming, not about numbers themselves.
Number itself is unproblematic; the paradox concerns the linguistic vehicle for referring to numbers, not their mathematical structure.
Related Experiments
Experiments engaged by an overlapping set of schools — likely to surface the same fault lines.
Further reading
- Russell, "Mathematical Logic as Based on the Theory of Types", *AJM* 30 (1908)
- Chaitin, "Information-Theoretic Limitations of Formal Systems" (1974)
Related Historical Debates
Debates that share dimensions and/or aligned schools with this experiment.
Personas Most Aligned With This Experiment
Ranked by total declared-influence weight in the schools that respond to this experiment.
Works Most Aligned With This Experiment
Ranked by total declared-influence weight in the schools that respond to this experiment.
Related Contemporary Dilemmas
Dilemmas that engage the same dimensions as this experiment.