Adding ultrafilters to Shelah's model for no nowhere dense ultrafilters

TL;DR

This paper introduces a forcing method to create models with specific ultrafilter properties, demonstrating that certain ultrafilters can exist independently of others, and satisfying the full Sacks Property.

Contribution

It presents a novel forcing construction that produces models with no nowhere dense ultrafilters while maintaining the full Sacks Property, and shows the independence of certain ultrafilter types.

Findings

Constructed a model with no nowhere dense ultrafilters satisfying the Sacks Property.

Produced a model with a $(2, eth 0)$-selective ultrafilter but no nowhere dense ultrafilters.

Proved that the existence of $(2, eth 0)$-selective ultrafilters does not imply the existence of nowhere dense ultrafilters.

Abstract

We exhibit a forcing for producing a model with no nowhere dense ultrafilters that satisfies the full Sacks Property. By interleaving this forcing with other forcing notions, a model containing a $(2, {\aleph}_{0})$-selective ultrafilter, but no nowhere dense ultrafilters is produced. It is thus proved that the existence of $(2, {\aleph}_{0})$-selective ultrafilters does not imply the existence of nowhere dense ultrafilters.