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Berry paradox
The Berry paradox is a self-referential paradox arising from an expression like "the smallest positive integer not definable in fewer than twelve words" (note that this defining phrase has fewer than twelve words). Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian library, who had suggested the more limited paradox arising from the expression "the first undefinable ordinal".
== The paradox ==
Consider the expression:
:"The smallest positive integer not definable in under sixty letters."
Since there are only twenty-six letters, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a ''smallest'' positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under sixty letters". This is the integer to which the above expression refers. The above expression is only fifty-seven letters long, so this integer is defined by an expression that is under eleven words long; it ''is'' definable in under sixty letters, and is ''not'' the smallest positive integer not definable in under sixty letters, and is ''not'' defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under sixty letters), there cannot be any integer defined by it.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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