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

TL;DR

This paper classifies the Borel complexity of isomorphism relations for complete theories of linear orders with unary predicates, revealing a spectrum from countably categorical to Borel complete, based on model-theoretic properties.

Contribution

It extends the classification of Borel complexity to theories of linear orders with unary predicates, providing a detailed model-theoretic characterization of each complexity case.

Findings

Th(A) is either $eth_0$-categorical or Borel complete.

Theories with unary predicates have finitely many models or are Borel equivalent to complex relations.

Precise conditions determine the Borel complexity case for each theory.

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.