It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on ${ \aleph}_{ \omega+1}$ which is generated by fewer than ${ 2}^{ { \aleph}_{ \omega+1}}$ sets.
Categories: math.LO