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

TL;DR

This paper investigates the complexity of isomorphism and bi-embeddability relations within countable models of L_{ω_1ω}-sentences, providing a near-complete characterization of which pairs of analytic equivalence relations can be realized in this framework.

Contribution

It offers an almost complete solution to the problem of realizing pairs of analytic equivalence relations as isomorphism and bi-embeddability on L_{ω_1ω}-elementary classes, under mild conditions.

Findings

Almost complete characterization of realizable pairs (E,F)

Conditions under which pairs of relations can be realized

Extension of previous work on analytic equivalence relations

Abstract

Given an L_{\omega_1 \omega}-elementary class C, that is the collection of the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphism and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal [LR05], in [FMR09] it was 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_C,\equiv_C), C some 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 problem: under very mild conditions on E and F, it is always possible to find such an L_{\omega_1 \omega}-elementary class C.