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.