Gruff Ultrafilters

TL;DR

This paper explores the existence of ultrafilters generated by perfect sets on the rational numbers, demonstrating their consistency under various set-theoretic assumptions, including diamond principles, cardinal invariants, and forcing models.

Contribution

It proves the consistent existence of gruff ultrafilters on  under multiple set-theoretic assumptions, expanding understanding of ultrafilter constructions.

Findings

Gruff ultrafilters can exist under a parametrized diamond principle

Existence is consistent when =

Such ultrafilters are consistent in the Random real model

Abstract

We investigate the question of whether $\mathbb Q$ carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality $\mathfrak d=\mathfrak c$, and in the Random real model. [Edit: replace "the Randor real model" with "the model obtained by adding $\omega_1$ Cohen reals to a model of $\mathsf{CH}$, and subsequently forcing with the Random algebra"; this is clarified in the corrigendum attached at the end of the paper.]