On the Existence of Nonregular Ultrafilters and the Cardinality of Ultrapowers
On the Existence of Nonregular Ultrafilters and the Cardinality of Ultrapowers
Assuming the consistency of huge cardinals, we prove that ${\omega _3}$ can carry an ultrafilter D such that ${\omega _1}^{{\omega _3}}/D$ has cardinality ${\omega _3}$. (Hence D is not $({\omega _3}, {\omega _1})$ regular.) Similarly ${\omega _2}$ can carry an ultrafilter D such that ${\omega ^{{\omega _2}}}/D$ has cardinality ${\omega …