Berry paradox

The Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words." Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928),^[1] a junior librarian at Oxford's Bodleian library, who had suggested the more limited paradox arising from the expression "the first undefinable ordinal".

The paradox

Consider the expression:

"The smallest positive integer not definable in under eleven words."

Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words. By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers. The above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under eleven words), there cannot be any integer defined by it.

Resolution

The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable." In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.^[2] To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.

The argument presented above that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression". This is counterfactual, as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.^[3]

More rigorously, this family of paradoxes can be resolved by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable₀ in less than eleven words' may be nameable₁ in less than eleven words under this scheme.^[4]

Formal analogues

Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results.

George Boolos (1989) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way. The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set {(n, k): n has a definition that is k symbols long} can be shown to be representable (using Gödel numbers). Then the proposition "m is the first number not definable in less than k symbols" can be formalized and shown to be a definition in the sense just stated.

Relationship with Kolmogorov complexity

It is possible to unambiguously define what is the minimal number of symbols required to describe a given string (given a specific description mechanism). In this context, the terms string and number may be used interchangeably, since a number is actually a string of symbols, i.e. an English word (like the word "eleven" used in the paradox) while, on the other hand, it is possible to refer to any word with a number, e.g. by the number of its position in a given dictionary, or by suitable encoding. Some long strings can be described exactly using fewer symbols than those required by their full representation, as is often experienced using data compression. The complexity of a given string is then defined as the minimal length that a description requires in order to (unambiguously) refer to the full representation of that string.

The Kolmogorov complexity is defined using formal languages, or Turing machines, that allow to avoid ambiguities about what string results from a given description. After defining that function, it can be proved that it cannot be computed. The proof by contradiction shows that if it were possible to compute the Kolmogorov complexity, then it would also be possible to systematically generate paradoxes similar to this one, i.e. descriptions shorter than what the complexity of the described string implies. That is to say, the definition of the Berry number is paradoxical because it is not actually possible to compute how many words are required to define a number, and we know that such computation is not feasible because of the paradox.

See also

• Definable number
• Busy beaver
• Richard's paradox
• Interesting number paradox

Notes

1. ^ Nicholas Griffin, The Cambridge companion to Bertrand Russell, p. 63.
2. ^ Russell and Whitehead (1927).
3. ^ French (1988) demonstrated that an infinite number of numbers could be uniquely described in the exact same words.
4. ^ Willard Quine (1976) Ways of Paradox. Harvard Univ. Press

References

• Charles H. Bennett (1979) "On Random and Hard-to-Describe Numbers." IBM Report RC7483.
• George Boolos (1989) "A new proof of the Gödel Incompleteness Theorem," Notices of the American Mathematical Society 36: 388–90; 676. Reprinted in his (1998) Logic, Logic, and Logic. Harvard Univ. Press: 383-88.
• Gregory Chaitin (1995) "The Berry Paradox,." Complexity 1: 26-30.
• French, James D. (1988) "The False Assumption Underlying Berry's Paradox," Journal of Symbolic Logic 53: 1220–1223.
• Bertrand Russell (1906) "Les paradoxes de la logique," Revue de métaphysique et de morale 14: 627–650
• Bertrand Russell and Alfred N. Whitehead (1927) Principia Mathematica. Cambridge University Press. 1962 partial paperback reissue goes up to *56.

External links

• Roosen-Runge, Peter H. (1997) "Berry's Paradox."
• Weisstein, Eric W., "Berry Paradox" from MathWorld.