# A small ultrafilter number at smaller cardinals

## TL;DR

This paper demonstrates the consistency of the existence of ultrafilters on certain uncountable sets generated by fewer sets than the maximum, relative to large cardinal assumptions.

## Contribution

It establishes the relative consistency of small ultrafilter numbers at smaller uncountable cardinals using large cardinal hypotheses.

## Key findings

- Existence of a uniform ultrafilter on the reals generated by fewer than continuum sets
- Existence of a uniform ultrafilter on ${
m
olinebreak} eth_{
olinebreak}
olimits_{
olinebreak}
olinebreak ext{+}
olinebreak 1$ with fewer than ${2}^{{
m
olinebreak} eth_{
olinebreak}
olimits_{
olinebreak}
olinebreak ext{+}}$ generators
- Consistency results relative to measurable and supercompact cardinals

## Abstract

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.