Model Theory Seminar

November 17
Scott Mutchnik, University of Illinois at Chicago
$\text{(N)SOP}_{2^{n+1}+1}$ Theories

Among the classical properties of unstable theories defined by Shelah, our understanding of the strict order hierarchy, $\text{NSOP}_n$, has remained relatively limited past $n = 4$ at the greatest. Methods originating from stability theory have given insight into the structure of stronger unstable classes, including simple and $\text{NSOP}_1$ theories. In particular, syntactic information about formulas in a first-order theory often corresponds to semantic information about independence in a theory's models, which generalizes phenomena such as linear independence in vector spaces and algebraic independence in algebraically closed fields. We discuss how the fine structure of this independence reveals exponential behavior within the strict order hierarchy, particularly at the levels $\text{SOP}_{2^{n+1}+1}$ for positive integers $n$. Our results suggest a potential theory of independence for $\text{NSOP}_n$ theories, for arbitrarily large values of $n$.