Paris–Harrington theorem

In mathematical logic, the Paris–Harrington theorem states that a certain claim in Ramsey theory, namely the strengthened finite Ramsey theorem, which is expressible in Peano arithmetic, is not provable in this system. That Ramsey-theoretic claim is, however, provable in slightly stronger systems.

This result has been described by some (such as the editor of the Handbook of Mathematical Logic in the references below) as the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem.