Cardinal Characteristics of Models of Set Theory

Abstract

We continue our investigation =of Shelah's interpretability orders ${⊴}_{\kappa }^{\ast }$ as well as the new orders ${⊴}_{\kappa }^{\times }$. In particular, we give streamlined proofs of the existence of minimal unstable, unsimple and nonlow theories in these orders, and we give a similar analysis of the hypergraph examples $T_{n,k}$ of Hrushovski. We also prove that if $\mathcal{B}$ is a complete Boolean algebra with the $\lambda$-c.c., then no nonprincipal ultrafilter on $\mathcal{U}$ ${\lambda }^{+}$-saturates any unsimple theory.

35 pages