The Non-Axiomatizability of the First-Order Theory of O-Minimality
University of Toronto
For a fixed language L, the first-order L-theory of o-minimality is the set of those L-sentences true in all o-minimal L-structures. It follows from a classical model-theoretic result that a model of this theory is either o-minimal or an elementary substructure of an ultraproduct of o-minimal L-structures. We call these structures pseudo-o-minimal, and note that they will generally not be o-minimal.
In this talk, I will discuss how the study of pseudo-o-minimality fits in to the ongoing project of classifying the tame weakenings of o-minimality. My main focus will be on the recent question of whether for certain fixed languages L, the first-order L-theory of o-minimality is recursively axiomatizable. I will show that it is not whenever L extends the language of ordered fields by at least one new predicate or function symbol. With the time remaining, I will outline some of what is known about the relative tameness of pseudo-o-minimal structures, and mention some open problems in the area.
Alex Rennet is a postdoc in the Mathematics department at the University of Toronto working under the supervision of Bill Weiss. His research focus right now is in o-minimality and in particular, ultraproducts of o-minimal structures. He received his Ph.D. in 2012 at the University of California at Berkeley, under the supervision of Thomas Scanlon.