Let $T$ be a complete theory without finite models. Prove that the following are equivalent:
- $T$ is $\omega$-categorical.
- $T$ has a countable model which is both atomic and saturated.
- All models of $T$ are $\omega$-saturated.
I already proved that if $T$ is $\omega$-categorical then $T$ has a countable model which is atomic - by a straigforward argument involving prime model -; nevertheless, the remaining saturated condition is missing. Also, I'm having problems on the rest of the exercise.
Any help would be really appreciated.
Hint: a theory is countably categorical iff there are finitely many $n$-types for each $n$ (this is (a part of) the Ryll-Nardzewski theorem). This implies that there are finitely many types over any finite set. See that this implies that every model is $\omega$-saturated.
You may want to use omitting types theorem.