Berry Paradox

Overview and Formulation

The Berry paradox is a self-referential semantic paradox involving apparently well-defined but ultimately problematic descriptions of natural numbers. A standard version uses the phrase “the smallest positive integer not nameable in fewer than nineteen English words.” This expression looks like a definite description that picks out a unique integer. Yet, if it does succeed in naming such an integer, it contradicts its own content: it names a number while asserting that no phrase of that length can name it.

The paradox arises from combining:

• a finite bound on expression length (for example, “fewer than nineteen words”),
• the idea of “nameable” or “definable” numbers in a natural language such as English,
• and self-reference: talking about what can be named “in fewer than N words” using a phrase that itself respects that bound.

By a simple counting argument, there are only finitely many English phrases of fewer than N words, so at most that many integers can be named using such phrases. Since there are infinitely many positive integers, there must be numbers that cannot be named in fewer than N words. Among them, there is a least such integer. But the phrase “the smallest positive integer not nameable in fewer than N words” appears to name that very “unnameable” integer, generating the paradox.

Historical Background

The paradox is named after the English librarian and amateur mathematician G. G. Berry. According to Bertrand Russell, Berry discussed examples of numbers defined by complicated English descriptions in correspondence and conversation. Russell introduced and popularized the paradox in his 1908 paper Mathematical Logic as Based on the Theory of Types and in subsequent writings.

The Berry paradox belongs to a family of early twentieth-century logical and semantic puzzles, alongside:

• Richard’s paradox, concerning definability of real numbers in language,
• Russell’s paradox, about sets that do or do not contain themselves,
• and the liar paradox, based on self-referential truth claims.

These paradoxes played an important role in motivating new approaches to the foundations of mathematics, including type theory, axiomatic set theory, and later developments in formal semantics and the theory of computation.

Philosophical and Logical Significance

The Berry paradox raises questions about definition, reference, and meaning in natural languages, and about the formal treatment of these notions.

1. Vagueness of “nameable” and “definable”

   The paradox hinges on an informal notion of “nameable in English in fewer than N words.” English is not precisely specified: boundaries between synonymous expressions, abbreviations, and technical vocabulary are unclear. Critics argue that once “nameable” is informally understood, the paradox does not yield a strict contradiction so much as it exposes the imprecision and context-dependence of natural language.

2. Self-reference and semantic closure

   The construction uses language that is, in a sense, semantically closed: English can talk about English phrases and about what can be named in English. This facilitates a form of self-reference akin to that in the liar paradox. Philosophers have used the Berry paradox to illustrate how self-referential characterizations of expressions—particularly those involving length, complexity, or definability—can generate contradictions or paradox-like phenomena.

3. Connection to definability and arithmetic

   In more formal settings, the Berry paradox is related to the idea of arithmetical definability. Instead of “English phrases,” one can consider formulas of a formal language (such as the language of arithmetic) and investigate which numbers they define. This leads to questions about:

   • whether every number that can be intuitively described can be formally defined, and
   • whether talk of “the smallest number not definable by a formula of length less than N” can be made precise without contradiction.

   These concerns intersect with model theory and descriptive set theory, which study formal notions of definability and their limitations.

4. Relation to Kolmogorov complexity and information theory

   A formally sharpened analogue of the Berry paradox appears in discussions of Kolmogorov complexity, the measure of the shortest description (e.g., program) that outputs a given object. Informally, one might speak of:

   "

   “the least integer that cannot be specified by a description shorter than M bits.”

   A parallel tension emerges: if such a number is designated by that phrase, then it has a short description, contrary to the stipulation. While modern theory avoids outright contradiction by defining complexity within a precisely delimited formal framework, the intuitive pull of the Berry paradox illuminates the limits of compressibility and description.

Responses and Resolutions

Philosophers and logicians have proposed several strategies for handling the Berry paradox. None is universally accepted as the single correct diagnosis, but together they show how the apparent contradiction can be defused.

1. Restricting or regimenting language

   One common response is to insist that expressions such as “nameable in fewer than N English words” are ill-formed or insufficiently precise in everyday language. On this view, the paradox dissolves once one works in a formalized language with:

   • a fixed alphabet and syntax,
   • a precise length measure for formulas,
   • and a strict definition of what counts as “defining” or “naming” a number.

   When such conditions are imposed, some of the moves needed to generate the paradox become inexpressible or blocked: for instance, the language might be unable to transparently speak about the definability of numbers within the same language without moving to a higher-level metalanguage.

2. Type-theoretic and hierarchical approaches

   Inspired by Russell’s own foundational work, some approaches introduce hierarchies of languages (object language vs. metalanguage) or type restrictions to prevent mischievous self-reference. Under such frameworks, phrases that quantify over “all English phrases of fewer than N words” and then attempt to be among those phrases are treated as illegitimate or as belonging to a higher type. The paradox is thereby avoided, though some see this as limiting the expressive power of the language.

3. Semantic and pragmatic analyses

   Some theorists suggest that the paradox exploits subtle features of meaning, context, and use in natural language. They argue that the phrase “not nameable in fewer than N words” shifts meaning once the very act of using it is considered, or that it relies on indexical or context-relative aspects of “nameable” that cannot be held fixed while performing the paradoxical reasoning. On these views, the argument shows more about the complexity of natural-language semantics than about mathematics itself.

4. Connections to incompleteness and incomputability

   Others see the Berry paradox as philosophically connected to Gödel’s incompleteness theorems and various diagonal arguments. Although not identical in form, Berry-type reasoning underscores that any sufficiently expressive system that can represent its own syntax or notions of definability risks generating statements that refer indirectly to their own (un)definability. In this interpretive tradition, the paradox is not simply dismissed, but is taken as an informal precursor to more formally robust theorems about the limits of formal systems.

Because of these divergent interpretations, the Berry paradox is often classified as “controversial” in terms of its status: it is not a straightforward logical contradiction in a fully specified formal system, but rather a philosophically charged puzzle about the interplay between language, reference, and mathematical description. It remains an important example in discussions of semantic paradoxes, definability, and the philosophy of mathematics.