Given an L_{ω_1 ω}-elementary class C, that is, the collection of the countable models of some L_{ω_1 ω}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on 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_C, \equiv_C), C some L_{ω_1 ω}-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 L_{ω_1 ω}-elementary class C.
On the complexity of the relations of isomorphism and bi-embeddability
MOTTO ROS, Luca
2012-01-01
Abstract
Given an L_{ω_1 ω}-elementary class C, that is, the collection of the countable models of some L_{ω_1 ω}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on 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_C, \equiv_C), C some L_{ω_1 ω}-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 L_{ω_1 ω}-elementary class C.
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