Definition
Berry's Paradox
Berry’s paradox is a semantic paradox of unrestricted, self-applicable definability.
Fix a finite English vocabulary containing the words below, and let
denote the positive natural numbers: Finitely many short descriptions make
finite; infinitely many positive integers make its complement non-empty. Taking
gives “the least positive integer not definable in under eleven words”. By construction, , hence . Yet this ten-word description appears to specify uniquely, giving : a contradiction. The problem is unrestricted self-applicable definability, not arithmetic: a description about a precise language’s definability need not be an admissible definition within that language.