## Doctoral Dissertations

#### Date of Award

8-2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Degree Program

Mathematics

#### University

Graduate College of the University of Illinois at Chicago

#### First Advisor

David E. Marker, Ph.D.

#### Keywords

Model Theory, Descriptive Set Theory, O-minimal, Borel Complete, Complexity of Isomorphism

#### Subject Categories

Logic and Foundations | Mathematics

#### Abstract

In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories (1). She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory *T* if *T* has fewer than 2K ° countable models. Friedman and Stanley have shown in (2) that several elementary classes are Borel complete. This work addresses the class of countable models of an o-minimal theory *T* when *T* has 2N ° countable models, including conditions for when this class is Borel complete. The main result is as follows.

**Theorem 1.** Let T be an o-minimal theory in a countable language having 2N ° countable models. Either

*i. For every finite set A, every p(x) E S _{1} (A) is simple, and isomorphism on the class of countable models of T is ∏^{0}_{3} (and is, in fact, equivalence of countable sets of reals); or*

*ii. For some finite set A, some p(x) E S _{1} (A) is non-simple, and there is a finite set B D A such that the class of countable models of T over B is Borel complete.*

#### Recommended Citation

Sahota, D. S. (2013). *Borel complexity of the isomorphism relation for O-minimal theories* (Doctoral dissertation). Retrieved from http://digitalcommons.imsa.edu/alumni_dissertations/12/