A Small Ultrafilter Number at Every Singular Cardinal

TL;DR

This paper constructs a model with a small ultrafilter number at a specific singular cardinal using advanced forcing techniques, while maintaining certain large cardinal properties and GCH conditions.

Contribution

It introduces a novel forcing method that preserves inaccessibility and controls ultrafilter numbers at singular cardinals.

Findings

Small ultrafilter number at leph_{\u03a9_1}

Model with leph_{\u03a9_1} as least inaccessible cardinal

GCH holds at regulars, SCH fails at singulars

Abstract

We obtain a small ultrafilter number at $\aleph_{\omega_1}$. Moreover, we develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal $\kappa$ inaccessible. We apply this forcing to construct a model where $\kappa$ is the least inaccessible and $V_\kappa$ is a model of GCH at regulars, failures of SCH at singulars, and the ultrafilter numbers at all singulars are small.