The Paris-Harrington theorem is a strengthening of the finite Ramsey's theorem by requiring that the homogeneous set be large enough so that card H>=min H. Clearly, the statement can be expressed in the first-order language of arithmetic. It is easily provable in the second-order arithmetic, but is unprovable in first-order Peano arithmetic.

Gödel's first incompleteness theorem | Gödel's second incompleteness theorem | Peano arithmetic | Robertson-Seymour theorem