A computability theoretic equivalent to Vaughts conjecture

We prove that, for every theory $T$ which is given by an ${mathcal L}_{omega_1,omega}$ sentence, $T$ has less than $2^{aleph_0}$ many countable models if and only if we have that, for every $Xin 2^omega$ 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 Vaughts conjecture.

A fixed point for the jump operator on structures

Assuming that $0^#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $mathcal A$ such that [ Sp({mathcal A}) = {{bf x}:{bf x}in Sp ({mathcal A})}, ] where $Sp ({mathcal A})$ is the set of Turing degrees which compute a copy of $mathcal A$. It turns out that, more interesting than the result itself, is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.