  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.
