The Diversity of Minimal Cofinal Extensions

Abstract

Fix a countable nonstandard model $\mathcal{M}$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $\mathcal{N}\succ \mathcal{M}$ that are allowed, we still find that there are $2^{{\aleph }_0}$ possible theories of $(\mathcal{N},M)$ for such $\mathcal{N}$'s.