Some Dichotomy Theorems for Isomorphism Relations of Countable Models

Abstract

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an $L_{\omega 1\omega}$-interpretability theory. We also give a survey of this theory.