The ultrafilter number for singular cardinals

TL;DR

This paper demonstrates the consistency of having a singular cardinal with a small ultrafilter number while simultaneously having a very large power set, showing independence results in set theory.

Contribution

It establishes the possibility of a singular cardinal with a small ultrafilter number and an arbitrarily large power set, expanding understanding of cardinal characteristics.

Findings

Consistency of small ultrafilter number at singular cardinals

Arbitrarily large power set of the singular cardinal

Advances in set-theoretic independence results

Abstract

We prove the consistency of a singular cardinal $\lambda$ with small value of the ultrafilter number $u_\lambda$, and arbitrarily large value of $2^\lambda$.