Monotonicity of the ultrafilter number function

TL;DR

This paper explores whether the ultrafilter number function on cardinals is monotone, demonstrating that non-monotonicity can occur with large cardinal assumptions, but also identifying conditions where monotonicity is preserved.

Contribution

It shows that the ultrafilter number function can fail to be monotone, with the failure requiring large cardinal strength, and identifies specific cases where monotonicity still holds.

Findings

Monotonicity can fail for the ultrafilter number function.

Failure of monotonicity requires large cardinal assumptions.

Monotonicity holds at certain singular cardinals, such as those with countable cofinality.

Abstract

We investigate whether the ultrafilter number function $\kappa \mapsto \mathfrak{u}(\kappa)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(\lambda) \le \mathfrak{u}(\kappa)$ holds for all cardinals $\lambda < \kappa$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $\kappa$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak{u}(\kappa) \le \mathfrak{u}(\kappa^+)$ holds.