Indestructibility and the linearity of the Mitchell ordering

期刊论文
发表日期：

2024年2月13日

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Arthur W. Apter

摘要

Suppose that $\kappa $ $κ$ is indestructibly supercompact and there is a measurable cardinal $\lambda > \kappa $ $λ > κ$ . It then follows that $A_0 = \{\delta < \kappa \mid \delta $ $A_{0} = {δ < κ ∣ δ$ is a measurable cardinal and the Mitchell ordering of normal measures over $\delta $ $δ$ is nonlinear $\}$ $}$ is unbounded in $\kappa $ $κ$ . If the Mitchell ordering of normal measures over $\lambda $ $λ$ is also linear, then by reflection (and without any use of indestructibility), $A_1= \{\delta < \kappa \mid \delta $ $A_{1} = {δ < κ ∣ δ$ is a measurable cardinal and the Mitchell ordering of normal measures over $\delta $ $δ$ is linear $\}$ $}$ is unbounded in $\kappa $ $κ$ as well. The large cardinal hypothesis on $\lambda $ $λ$ is necessary. We demonstrate this by constructing via forcing two models in which $\kappa $ $κ$ is supercompact and $\kappa $ $κ$ exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that $A_0$ $A_{0}$ is unbounded in $\kappa $ $κ$ if $\lambda > \kappa $ $λ > κ$ is measurable. In one of these models, for every measurable cardinal $\delta $ $δ$ , the Mitchell ordering of normal measures over $\delta $ $δ$ is linear. In the other of these models, for every measurable cardinal $\delta $ $δ$ , the Mitchell ordering of normal measures over $\delta $ $δ$ is nonlinear.