This Corollary states that a complete theory $T$ in a countable language with infinite models is $\omega$-stable when it is $\lambda$-categorical for some $\lambda\geq\aleph_1$.
I am confused why the author jumped through the case that $|S^M_n(A)|<\aleph_0$. I think we also need to prove this case does not exist to say that $|S^M_n(A)|$ is exactly $\aleph_0$.
So my question: how should we prove $|S^M_n(A)|\geq\aleph_0$?
P.S. Some other authors define $\kappa$-stable to be $|S^M_n(A)|\leq\kappa$, what's the relation between this definition and $|S^M_n(A)|=\kappa$?
The key observation is that $S^M_n(A)$ contains the type $tp^M(a/A)$ for all $a \in A$. So $|S^M_n(A)|$ has at least the same cardinality as $A$.
Then the subtlety is that Marker requires $|A| = \aleph_0$ in his definition, while other authors require just $|A| \leq \aleph_0$.