On the complexity of the relations of isomorphism and bi-embeddability

Abstract:

Given an $\mathcal {L}_{\omega _1\omega }$-elementary class $\mathcal {C}$, that is, the collection of the countable models of some $\mathcal {L}_{\omega _1 \omega }$-sentence, denote by $\cong _{\mathcal {C}}$ and $\equiv _{\mathcal {C}}$ the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on $\mathcal {C}$. Generalizing some questions of A. Louveau and C. Rosendal, in a paper by S. Friedman and L. Motto Ros they proposed the problem of determining which pairs of analytic equivalence relations $(E,F)$ can be realized (up to Borel bireducibility) as pairs of the form $(\cong _{\mathcal {C}}, \equiv _{\mathcal {C}})$, $\mathcal {C}$ some $\mathcal {L}_{\omega _1\omega }$-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such a problem: under very mild conditions on $E$ and $F$, it is always possible to find such an $\mathcal {L}_{\omega _1\omega }$-elementary class $\mathcal {C}$.