The Borel Complexity of Isomorphism for Some First-order Theories

Title of dissertation: THE BOREL COMPLEXITY OF ISOMORPHISM FOR SOME FIRST-ORDER THEORIES Richard Rast, Doctor of Philosophy, 2016 Dissertation directed by: Professor Michael C. Laskowski Department of Mathematics In this work we consider several instances of the following problem: “how complicated can the isomorphism relation for countable models be?” Using the Borel reducibility framework from [4], we investigate this question with regard to the space of countable models of particular complete first-order theories. We also investigate to what extent this complexity is mirrored in the number of back-andforth inequivalent models of the theory, denoted I∞ω(T ). We consider this question for two large and related classes of theories. First, we consider o-minimal theories, showing that if T is o-minimal, then ∼=T is either Borel complete or Borel. Further, if it is Borel, then it is exactly equivalent to one of the following: ∼=1, ∼=2, or (36,=), with a, b ∈ ω. All values are possible, and we characterize exactly when each possibility occurs. Further, in all cases Borel completeness implies λ-Borel completeness for all λ. Much of this portion appeared in [21] and extends work from [25], which itself builds upon [15]. Second, we consider colored linear orders, which are (complete theories of) a linear order expanded by countably many unary predicates. We discover the same characterization as with o-minimal theories, taking the same values, with the exception that all finite values are possible except two. We characterize exactly when each possibility occurs, which is similar to the o-minimal case. Additionally, we extend Schirrman’s theorem from [26], showing that if the language is finite, then T is א0-categorical or Borel complete. As before, in all cases Borel completeness implies λ-Borel completeness for all λ. This work appeared in [20] and builds heavily on [24]. The Borel Complexity of Isomorphism for Some First-Order Theories