The Borel Complexity of Isomorphism for Complete Theories of Linear Orders With Unary Predicates

Abstract: We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise model-theoretic conditions indicated which case occurs. This complements work on o-minimal theories where analogous results were shown. A large portion of the machinery under the proof is based on work by Matatyahu Rubin in "Theories of Linear Order," where it was shown that such a theory satisfies Vaught's Conjecture and cannot have precisely $\aleph_0$ countable models.