Date of Award

8-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

College

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 S1 (A) is simple, and isomorphism on the class of countable models of T is ∏03 (and is, in fact, equivalence of countable sets of reals); or

ii. For some finite set A, some p(x) E S1 (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.

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