Pseudosaturation and the Interpretability Orders

Abstract

We streamline treatments of the interpretability orders ${⊴}_{\kappa }^{\ast }$ 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 $T_0{⊴}_1^{\ast }{T}_1$ if and only if $T_0{⊴}_{{\aleph }_1}^{\ast }{T}_1$. We thus deduce that simplicity is a dividing line in ${⊴}_{{\aleph }_1}^{\ast }$, and that consistently, $SO{P}_2$ characterizes maximality in ${⊴}_{{\aleph }_1}^{\ast }$; previously these results were only known for ${⊴}_1^{\ast }$.

42 pages