Berry paradox

The Berry paradox arises when one attempts to evaluate expressions like the following:

Solutions

It is generally accepted that the Berry paradox and its ilk hang on underdetermined language. That is, "not nameable in fewer than n words" is not well-defined, for several reasons. First, the number of words in which a thing is nameable is surely relative to the language in which it is named. So we might attempt to prop up the paradox as follows: "The smallest number not nameable in English in fewer than thirteen words" (adjusting word counts as the length of the expression requires). But this is not really enough either, because we can name any number in as few words as we want, just by stipulation. Pick out whatever number you want using as long an expression as you need, and declare, "I shall henceforth call that number Jones." Now you can ask whether Jones is odd, or even, or prime, and name it using only one word.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We can invent new words in this way whenever we please. Yet it is a well known fact that there are more numbers than there are possible names (of reasonable syllabic length) in any language, so given any vocabulary there must be some number it cannot name in less than n words, for any n.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The real problem, then, is that the paradox must be formulated relative to a fixed vocabulary. So we might say, "The smallest number that cannot be named, by the totality of English that existed by the end of December 31, 1999, in fewer than twenty-eight words." (counting 31, 1999, twenty, and eight each as a single word.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

However, it was shown by Tarski that certain predicates, such as the truth-predicate for a language, can be formulated coherently only in a richer language than the one they apply to, a metalanguage. That is, the above predicate can only exist without contradiction in a language other than "the totality of English that existed by December 31, 1999." So the "paradox" expression is not in fact a counterexample to the condition it states.

Related Topics:
Tarski - Metalanguage

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(A minor quibble is that "the smallest number not nameable in fewer than eleven words" is not a name at all but a description. The paradox easily accommodates this with, "the smallest number not denotable by any expression of fewer than fourteen words." The solution is similar.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

However, Berry's paradox can be forced into a formal language, specifically that of mathematical logic, provided that enough axioms are assumed that their model is sufficiently strong to carry out ordinary arithmetic. Boolos used a specific formalization to provide an alternate proof of Godel's Incompleteness Theorem. The basic idea of the proof is that a proposition that holds of x iff x=n for some natural number n can be called a "name" for x, and that the set {(n,k): the natural number n has a name that is k symbols long} can be shown to be representable (using Gödel numbers). Then the proposition "m is the first number not nameable in under k symbols" can be formalized and shown to be a name.

Related Topics:
Godel's Incompleteness Theorem - Proposition

~ ~ ~ ~ ~ ~ ~ ~ ~ ~