Abstract

Idea of a Foundation for Mathematics: I will review some of the history of foundations, starting from Frege through to the Hilbert Program, leading up to the Incompleteness Theorems of 1931 due to Kurt Gödel. I will discuss my own approach to foundations at the end, a “local foundations” point of view.

Canonical Ramsey Theory: The infinite Ramsey’s Theorem states that whenever all pairs of natural numbers are colored by finitely many colors, there is an infinite set on which all pairs have the same color. When one wishes to use infinitely many colors, in other words an equivalence relation, it is not always possible to find an infinite set on which all pairs have the same color. However, a breakthrough of Erdos and Rado show that there is always an infinite set on which the equivalence relation is one of only four canonical types. We will discuss this canonical Ramsey theorem and some of our related work finding canonical equivalence relations on other classes of finite structures with the Ramsey property, as well as applications in set theory.