Michael Wheeler Nonstandard Analysis: A Talk about Almost Nothing
Nonstandard analysis was originally used to answer the longstanding controversy in calculus as to whether the use of infinitesimals could be given a rigorous foundation. However, the nonstandard methods have since been applied to several other fields (notably infinite Ramsey theory and topological dynamics) to significantly reduce the complexity of certain (rather long) proofs. While we will not cover these examples explicitly, the goal will be to provide enough of the machinery of nonstandard analysis to be able to see how nonstandard methods and interpretations can be applied to a variety of disciplines. In particular, we will describe what a nonstandard framework is, discuss some properties of frameworks that are enlargements, and give a quick proof of the Stone representation theorem for Boolean algebras using nonstandard methods. Time permitting, we will finish with a brief listing of some interesting properties of the hyperreals, including how to think about germs of functions using monads.