A separation theorem for simple theories

TL;DR

This paper introduces a new model-theoretic tool called shearing to analyze complexity differences in simple theories, leading to a separation theorem that distinguishes c-superstability from c-unsuperstability.

Contribution

It generalizes dividing to shearing depending on a context c, and proves a separation theorem using generalized Ehrenfeucht-Mostowski models, advancing the understanding of complexity in simple theories.

Findings

Shearing generalizes dividing in simple theories.

A separation theorem distinguishes c-superstability from c-unsuperstability.

Models can be constructed with specific saturation properties based on theory complexity.

Abstract

This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a syntactical notion, which includes supersimplicity as a special case. We prove a separation theorem showing that for any countable context c and any two theories $T_1$, $T_2$ such that $T_1$ is c-superstable and $T_2$ is c-unsuperstable, and for arbitrarily large $\mu$, it is possible to build models of any theory interpreting both $T_1$ and $T_2$ whose restriction to $\tau(T_1)$ is $\mu$-saturated and whose restriction to $\tau(T_2)$ is not $\aleph_1$-saturated. (This suggests "c-superstable" is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler's order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes.