Natural Classes of Countable Models

In the previous two chapters we have developed the theory of countable models in some depth from the perspectives of both mathematical logic and descriptive set theory. For invariant Borel classes with Borel isomorphism relations we have obtained satisfactory characterizations of the models. These also led to powerful results about Borel S∞-orbit equivalence relations. However, when natural classes of countable models are considered the isomorphism relations turn out to be mostly non-Borel, and often correspond to the universal S∞-orbit equivalence relation. In this chapter we consider various natural classes of countable models and substantiate this observation. Historically these natural invariant Borel classes are not only examples to interpret the theoretical results, but also an integral part of the countable model theory and source of inspirations for further theoretical tools. In this chapter we will focus on countable graphs, trees, linear orderings, and groups. Similar work has been done for countable lattices, fields, and Boolean algebras.