Many Normal Measures

Abstract

We characterize the situation of having at least $(2^{\kappa }{)}^{+}$-many normal ultrafilters on a measurable cardinal $\kappa$. We also show that if $\kappa$ is a compact cardinal, then $\kappa$ carries $(2^{\kappa }{)}^{+}$-many $\kappa$-complete ultrafilters, each of which extends the club filter on $\kappa$.