The Paradox
Consider the expression:
- "The smallest positive integer not definable in under eleven words."
Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words. By the well ordering principle, 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 eleven words". This is the integer to which the above expression refers. The above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, 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 eleven words), there cannot be any integer defined by it.
Read more about this topic: Berry Paradox
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