Abstract

We study a generalization of the splitting number $\mathfrak{s}$ to uncountable cardinals. We prove that $\mathfrak{s} > \kappa^+$ for a regular uncountable cardinal $\kappa$ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s} > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.