Abstract
We prove that, for every theory T which is given by an Lω1,ω sentence, T has less than 2ℵ0 many countable models if and only if we have that, for every X∈2ω on a cone of Turing degrees, every X-hyperarithmetic model of T has an X-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught’s conjecture.
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