arithmetic. It is provable in second-order arithmetic (or the far stronger Zermelo–Fraenkel set theory) and so is true in the standard model. The smallest number N that satisfies the strengthened finite Ramsey theorem is then a computable function of n, m, k, but grows extremely fast. In particular it is not primitive recursive, but it is also far larger than standard examples of non-primitive recursive functions such as the Ackermann function. Its growth is so large that Peano arithmetic cannot prove it is defined everywhere, although Peano arithmetic easily proves that the Ackermann function is well defined. See also Goodstein's theorem Kanamori–McAloon theorem Kruskal's tree theorem References mathworld entry External links A brief introduction to unprovability (contains a proof of the Paris–Harrington theorem) by Andrey Bovykin Independence results Theorems in the foundations of mathematics