Pseudosaturation and the Interpretability Orders

Abstract

We streamline treatments of the interpretability orders $\trianglelefteq^*_\kappa$ of Shelah, the key new notion being that of pseudosaturation. Extending work of Malliaris and Shelah, we classify the interpretability orders on the stable theories. As a further application, we prove that for all countable theories $T_0, T_1$, if $T_1$ is unsupersimple, then $T_0 \trianglelefteq^*_1 T_1$ if and only if $T_0 \trianglelefteq^*_{\aleph_1} T_1$. We thus deduce that simplicity is a dividing line in $\trianglelefteq^*_{\aleph_1}$, and that consistently, $SOP_2$ characterizes maximality in $\trianglelefteq^*_{\aleph_1}$; previously these results were only known for $\trianglelefteq^*_1$.