There are an infinite number of positive integers. There are only a finite number of words in the English language at any one time, and so there are only a finite number of phrases of twenty-four syllables or less. Therefore, the majority of integers must not be specifiable in less than twenty-five syllables.
If we have a set of positive integers, there must be a least member. Therefore, there must be a smallest positive integer not specifiable in less than twenty-five syllables.
But doesn't the specification "the smallest positive integer not specifiable in less than twenty-five syllables" have less than twenty-five syllables?
Therefore, the smallest positive integer not specifiable in less than twenty-five syllables can in fact be specified in less than twenty-five syllables. Thus, the least member of the set isn't really a member of the set after all. Therefore there is no smallest member. Without a smallest member, the entire set collapses.
So does that mean that there are no numbers not specifiable in less than twenty-five syllables? Of course not, or we could only have a finite number of integers. So what's wrong? And is that Gödel I see lurking round the corner again...?
This paradox is known as Berry's Paradox, and was published by Russell in the begining of the last century.
[It has been argued in the comments that the phrase "The smallest positive integer not specifiable..." is not a definition. While I think that this is an interesting angle, I'd argue that "The smallest integer not specifiable in less than two syllables" quite unambiguously defines seven, and so we should expect that this idea could be inducted to twenty-five syllables.] |