A natural example of an unprovable statement.

One possible attitude to Gödel's theorem is that the examples he gives of unprovable statements are all rather artificial, and unlikely to occur in `normal' mathematics. A famous result of Paris and Harrington makes this view hard to sustain. They exhibit a simple, natural and entirely finite Ramsey-theoretic statement which can be deduced easily from the infinite version of Ramsey's theorem. They also show that this use of the infinite is in a certain sense necessary: the result does not follow from the Peano axioms alone.