Model-theoretic applications of cofinality spectrum problems

Abstract

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is $\lambda$-saturated iff it has cofinality $\geq \lambda$ and the underlying order has no $(\kappa, \kappa)$-cuts for regular $\kappa < \lambda$. Second, assuming instances of GCH, we prove that $SOP_2$ characterizes maximality in the interpretability order $\trianglelefteq^*$, settling a prior conjecture and proving that $SOP_2$ is a real dividing line. Third, we establish the beginnings of a structure theory for $NSOP_2$, proving that $NSOP_2$ can be characterized by the existence of few inconsistent higher formulas. In the course of the paper, we show that $\mathfrak{p}_s = \mathfrak{t}_s$ in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.