The complexity of isomorphism for complete theories of linear orders with unary predicates

Abstract

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of $$\text {Th}(A)$$Th(A) up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite (in particular, if there are no unary predicates), then the theory is $$\aleph _0$$ź0-categorical or Borel complete; this generalizes a theorem due to Schirmann (Theories des ordres totaux et relations dequivalence. Master's thesis, Universite de Paris VII, 1997).