Because of Godel's Incompleteness Theorems, there must exist certain statements about the numbers which have no proof via the Peano axioms. This includes The Strengthened Ramsey Theorem.
However, this leads me to wonder if there are any statements in which we know there cannot be a proof showing the statement to be true using the Peano axioms, but there might be a proof showing the statement to be false (i.e. a counterexample), although we haven't found one. Do such statements exist?