A computability theoretic equivalent to Vaught's conjecture

We prove that, for every theory $T$ which is given by an ${\mathcal L}_{ω_1,ω}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^ω$ 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.

No reviews yet.

No comments yet.