Applications of Vaught Sentences and the Covering Theorem

Abstract

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem $0.1^\ast$ below) to derive several known and a few new results related to the logic $L_{\omega_1\omega}$. Among others, we prove that if every countable model in a $PC_{\omega_1\omega}$ class has only countably many automorphisms, then the class has either $\leq\aleph_0$ or exactly $2^{\aleph_0}$ nonisomorphic countable members (cf. Theorem $4.3^\ast$) and that the class of countable saturated structures of a sufficiently large countable similarity type is not $PC_{\omega_1\omega}$ among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks ($\s 7$).