The Diversity of Minimal Cofinal Extensions

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.