On the prime model property

Abstract:

Assume $T$ is superstable, $\Phi (x)$ is a formula over $\emptyset$, $Q=\Phi (M^*)$ is countable and $K_Q=\{M: M$ is countable and $\Phi (M)=Q\}$. We investigate models in $K_Q$ assuming $K_Q$ has the prime model property. We prove some corollaries on the number of models in $K_Q$. We show an example of an $\omega$-stable $T$ and $Q$ with $K_Q$ having exactly 3 models.