All the statements "P"("n") are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each "n", Peano arithmetic can prove that "P"("n") is true, but Peano arithmetic cannot prove the statement ""P"("n") is true for all "n"". Moreover the length of the shortest proof of "P"("n") in Peano arithmetic grows phenomenally fast as a function of "n", far faster than any primitive recursive function or the Ackermann function for example. The least "m" for which "P"("n") holds similarly grows extremely quickly with "n".