Applications of Vaught sentences and the covering theorem
Journal of Symbolic Logic 41 (1):171-187 (1976)
| 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 * below) to derive several known and a few new results related to the logic L ω 1 ω . Among others, we prove that if every countable model in a PC ω 1 ω class has only countably many automorphisms, then the class has either ≤ℵ 0 or exactly 2 ℵ 0 nonisomorphic countable members (cf. Theorem 4.3 * ) and that the class of countable saturated structures of a sufficiently large countable similarity type is not PC ω 1 ω among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks ( $\s 7$ ) | |||||||||
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