In 1931, Gödel proved his famous incompleteness theorem which implies that there exist true formulas in the language of arithmetic which are not provable from the Peano axioms (PA). However, the formula constructed in Gödel's proof involves the coding of formula and other logical apparatus and is therefore considered to be "unnatural." This leaves open the possibility of a "natural" mathematical statement which can be stated in the language of arithmetic but not provable in PA. The first such example did not appear until 1977 when Paris and Harrington showed that a slight strengthening of the finite Ramsey theorem (which we will abbreviate PH) is not provable in PA.

In the first talk, we will show that PH is a theorem of ZFC since it can be deduced from the infinite Ramsey theorem. We will then show that PA+PH implies another combinatorial principle which we will denote RP. In the second talk, we will use RP to show that given any nonstandard model M of PA+PH, there is an initial segment N of M which is a model of PA+(¬PH). From this, it follows that PH is not provable in PA.