Berry's paradox is a paradox, devised by G. G. Berry of the Bodleian Library at Oxford University in 1906, that involves statements of the form: "The smallest number not nameable in under ten words." At first sight, there doesn't seem anything particularly mysterious about this sentence. After all, there are only so many sentences that have less than ten words, and only a set S of these specify unique numbers; so there is clearly some number N that is the smallest integer not in S. The trouble is, the Berry sentence itself is a specification for that number in only nine words! Berry's paradox shows that the concept of nameability is inherently ambiguous and a dangerous concept to be used without qualification. A similar air of the paradoxical swirls around the notion of interesting numbers.

 

Reference

1. Chaitin, G. J. "The Berry Paradox." Complexity, 1: 26-30 (1995).