Indestructibility and the linearity of the Mitchell ordering

Abstract

Suppose that $\kappa$ is indestructibly supercompact and there is a measurable cardinal $\lambda > \kappa$. It then follows that $A_0 = \{\delta < \kappa \mid \delta$ is a measurable cardinal and the Mitchell ordering of normal measures over $\delta$ is nonlinear$\}$ is unbounded in $\kappa$. If the Mitchell ordering of normal measures over $\lambda$ is also linear, then by reflection (and without any use of indestructibility), $A_1= \{\delta < \kappa \mid \delta$ is a measurable cardinal and the Mitchell ordering of normal measures over $\delta$ is linear$\}$ is unbounded in $\kappa$ as well. The large cardinal hypothesis on $\lambda$ is necessary. We demonstrate this by constructing via forcing two models in which $\kappa$ is supercompact and $\kappa$ exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that $A_0$ is unbounded in $\kappa$ if $\lambda > \kappa$ is measurable. In one of these models, for every measurable cardinal $\delta$, the Mitchell ordering of normal measures over $\delta$ is linear. In the other of these models, for every measurable cardinal $\delta$, the Mitchell ordering of normal measures over $\delta$ is nonlinear.