Name the smallest number that can't be named in under twenty syllables
Consider this phrase: 'the smallest positive integer not nameable in fewer than twenty syllables.' Count its own syllables — there are fewer than twenty. But the phrase claims to name a number that, by definition, needs twenty or more syllables to name. So has this phrase actually named that number, or not?
Reveal the answer
Either answer breaks something: if the phrase does name the number, it has just named it in under twenty syllables, contradicting the number's own definition; if it doesn't, no contradiction arises, yet the phrase still seems to obviously refer to something. The paradox is credited to G. G. Berry, a junior librarian at Oxford's Bodleian Library, and was published by Bertrand Russell in 1908 as an example of the trouble self-reference causes for any formal notion of 'definability.'