Date of Award
Doctor of Philosophy (PhD)
Graduate College of the University of Illinois at Chicago
David E. Marker, Ph.D.
Model Theory, Descriptive Set Theory, O-minimal, Borel Complete, Complexity of Isomorphism
Logic and Foundations | Mathematics
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.
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/