Paradox of Berry

The Paradoxe of Berry was formulated by Bertrand Russell in 1906. One finds it in a article, published in French this same year, Revue of metaphysics and morals . Introduced Russell, in a discussion in connection with the Paradox of Richard, the “ smaller entirety not nommable in less than eighteen syllables which appears to be thus named in seventeen syllables”, and allots this paradoxical definition to a London librarian, G.G. Berry.

Always according to Russell, it is a simplification, which “has the merit not to exceed the finished numbers”, of the paradox of the “smaller ordinal indefinable which seems defined by the sentence even which announces that it is indefinable” (form probably due to Russell itself). These statements are included in the article of Russell of 1908 on the theory of the types.

Statement

“Smallest Whole nondescribable naturalness by an expression of fifteen words or less. ”

This number does it belong to the whole of the natural entireties describable by an expression of fifteen words or less?

Explanations

The natural entireties can be described by statements (in French) such as: “ten power hundred” or “the greatest prime number known at the twentieth century”. As the vocabulary available is finished (let us put that there are 200.000 French words), the statements of NR words cannot describe any more whole

200000^N

(and makes some much less; the majority of the “sentences” not wanting do anything of it to say, or not speaking about entireties).

The whole of the “describable natural integers by an expression of fifteen words or less” is thus finished; also there of many entireties out of this unit exists inevitably. Smallest of them is thus “the smallest nondescribable natural entirety by an expression of fifteen words or less”. But precisely, this statement which describes it perfectly, comprises only fifteen words.

One could also propose to create new words, but they are not of infinite number if one poses a limit with the number of letters: it would be enough to rewrite the statement with a limit of letters and not of words to circumvent this argument.

This paradox is very close to the Paradoxe of Richard (it is sometimes given besides under this name), of which it can be regarded as a finished alternative. Poincaré, which made a point of seeing the reason of the logical paradoxes in a handling without precautions of the infinite one, said, in connection with the paradox of Berry which precisely uses only finished concepts, “ they logicians tightened themselves the trap where they had fun to fall, and even they were obliged to pay attention well not to fall beside the trap ”.

One can also consider that it brings into play the same kind of questions as certain forms of the Paradoxe of the liar (the sentence who says itself which it is false). One usually solves it by formalizing the language, here that which makes it possible to describe the entireties, and by distinguishing it from the meta-language in which the sentence of Berry is stated which is not then paradoxical any more (see also the article on the Paradoxe of Richard).

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