Suppose that $T$ is uncountably categorical. By the Baldwin-Lachlan Theorem, we note that $I(T,\aleph_0)=1$ or $\aleph_0$.
Suppose that $I(T,\aleph_0)=\aleph_0$. Is it always the case that we get the nice embedding property that we see with models of $(\mathbb{N};s)$? Namely, there exists a prime model and the $non-prime$ models are linearly ordered by elementary embeddings? i.e. (1) there exists a $\mathfrak{A}$ such that $|\mathfrak{A}|=\aleph_0$ and for all $\mathfrak{B}$, $|\mathfrak{B}|=\aleph_0$, we have $\mathfrak{A} \prec \mathfrak{B}$. (2) There exists a linear order $X$ on the countable models such that for $\mathfrak{B}_i,\mathfrak{B}_j$, if $i < j$, then $\mathfrak{B}_i \prec \mathfrak{B}_j$?
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