A Classification of Nonstandard Models of Peano Arithmetic by Goodstein's Theorem presented by Dan Kaplan
Thursday, May 3, 2012
Abstract: In this paper I intend to outline a method for finding nonstandard models of Peano Arithmetic (PA) that satisfy Goodstein's theorem. Goodstein's Theorem is an interesting result because, though it is expressible completely in the language of number theory, it is nonetheless independent of the axioms of PA. I begin by rehearsing a proof of Goodstein's theorem, followed by a proof of its independence, developing the necessary tools to do so along the way. Finally, using indicator theory, I show how it is possible to classify the nonstandard models according to Goodstein's Theorem.
In the presentation I will mostly be focusing on rehearsing a proof of Goodstein's theorem and some independence results, gesturing at the end about how a classification would go. I am intending for it to be accessible to an undergraduate mathematics crowd.