Model-theoretic applications of cofinality spectrum problems

TL;DR

This paper uses cofinality spectrum problems to establish new model-theoretic results, including characterizations of saturation, interpretability order, and properties of NSOP2, advancing the understanding of model classification.

Contribution

It introduces novel applications of cofinality spectrum problems to prove key theorems in model theory, including characterizations of saturation and dividing lines like SOP2.

Findings

Models of Peano arithmetic are $ ext{λ}$-saturated iff they have cofinality ≥ λ and no $( ext{κ}, ext{κ})$-cuts for regular κ<λ.

$SOP_2$ characterizes maximality in interpretability order $ rianglelefteq^*$ under GCH assumptions.

$NSOP_2$ can be characterized by the existence of few inconsistent higher formulas.

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.