Berry paradox

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

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:The smallest positive integer not nameable in under eleven words.

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It is reasonable to assume that this is a specification for a number: after all, there are a finite number of sentences of fewer than eleven words, and some finite subset of them specify unique positive integers, so there is clearly some positive number that is the smallest integer not in that finite set.

Related Topics:
Finite - Positive - Integer

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But the Berry sentence itself is a specification for that number in only ten words!

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This is clearly paradoxical, and seems to indicate that "nameable in under eleven words" is not cleanly enough defined. Using programs or proofs of bounded lengths, one may in fact construct a rigorous version of the paradox; this has been done by Gregory Chaitin in order to produce an incompleteness theorem similar in spirit to Gödel's incompleteness theorem; see algorithmic information theory for an exposition.

Related Topics:
Paradoxical - Gregory Chaitin - Gödel's incompleteness theorem - Algorithmic information theory

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The Berry paradox was actually created by Bertrand Russell, who named it after G. G. Berry. Berry had provided the original idea in a letter to Russell about the less specific "the first ordinal that cannot be named in a finite number of words".

Related Topics:
Bertrand Russell - G. G. Berry - Ordinal

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Other versions of the Berry paradox exist as well, including the following:

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:* The smallest positive integer that requires more characters than there are in this sentence to be specified.

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:* The smallest positive integer not nameable in under one billion words.

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:* The first number not nameable in under ten words.

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