A separation theorem for simple theories

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 $\mathbf {c}$. This leads to defining $\mathbf {c}$-superstability, a syntactical notion, which includes supersimplicity as a special case. The main result is a separation theorem showing that for any countable context $\mathbf {c}$ and any two theories $T_1$, $T_2$ such that $T_1$ is $\mathbf {c}$-superstable and $T_2$ is $\mathbf {c}$-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 “$\mathbf {c}$-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.