Full Idea
Berry's Paradox refers to 'the least integer not namable in fewer than nineteen syllables' - a paradox because it has just been named in eighteen syllables.
Gist of Idea
Berry's Paradox finds a contradiction in the naming of huge numbers
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
Book Reference
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.71
A Reaction
Apparently George Boolos used this quirky idea as a basis for a new and more streamlined proof of Gödel's Theorem. Don't tell me you don't find that impressive.