Logic Seminar
Alex Rennet
University of Toronto
Non-Axiomatizability and Pseudo-O-Minimal Structures
Abstract: 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. For most languages, the latter kind of model will not in general be o-minimal; we call these structures pseudo-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, mention some open problems in the area, and discuss some potential applications.
Tuesday February 26, 2013 at 4:00 PM in SEO 427