We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.
We also investigate Borel isomorphisms between Borel structures.
"Borel structures and Borel theories." J. Symbolic Logic 76 (2) 461 - 476, June 2011. https://doi.org/10.2178/jsl/1305810759