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Peano Arithmetic is a first-order theory that describes all the notions we intuitively associate with working within the natural numbers: associative and distributive addition and multiplication, ordering, induction, etc. Although PA is quite powerful, Godel famously proved in 1931 that even consistent recursively enumerable theories containing PA could not, for example, prove or disprove their own consistency. The aim of this talk is to present two theorems, one by Goodstein, and the other by Kirby and Paris, which are also not provable in PA. Unlike Godel's, these theorems are number theoretic and combinatorial in nature, and as such can provide a more concrete feeling for the limits of PA. |