I would very much appreciate a hint as I have currently found myself stuck with exercise 13.2.2 in 'Introduction to Model Theory' by Philipp Rothmaler. Before relaying the problem I should perhaps also point out that in this text all theories are defined to be consistent as well as deductively closed.
The problem is as follows:
Suppose $T$ is a countable complete theory without finite models. Show that if $T$ has an (elementarily) prime model which is not minimal then $T$ has an atomic model of power $\aleph_1$.
Certainly, if $\mathfrak{N}$ is a prime model of $T$ then $\mathfrak{N}$ is unique up to isomorphism as well as atomic (and non-minimal by assumption). I guess the idea is to judiciously construct an uncountable atomic model out of countable atomic models and making use of the downwards Löwenheim-Skolem theorem but I am certainly missing something.
Best Answer
Hint: Build an elementary chain $(M_\alpha)_{\alpha<\aleph_1}$ of countable models, with $M_\alpha$ a proper elementary substructure of $M_\beta$ for all $\alpha<\beta$, and such that each $M_\alpha$ is isomorphic to the prime model $M$. Show that the union of this chain is atomic and has cardinality $\aleph_1$.
To handle the limit step of the transfinite construction, you will need to use the fact that a countable model is atomic if and only if it is prime, and countable prime models are unique up to isomorphism if they exist.