Abstract
The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle $a \in 2^\omega$, we can ask the same question relativized to $a$. A negative answer for every $a$ implies Vaught's Conjecture for $L_{\omega_1 \omega}$.
Citation
Howard Becker. "Isomorphism of computable structures and Vaught's Conjecture." J. Symbolic Logic 78 (4) 1328 - 1344, December 2013. https://doi.org/10.2178/jsl.7804180
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