A Classification of Nonstandard Models of Peano Arithmetic by Goodstein’s Theorem

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 that can classify the nonstandard models according to Goodstein's Theorem.