The Filter Dichotomy Principle Does not Imply the Semifilter Trichotomy Principle

TL;DR

This paper demonstrates that the filter dichotomy principle does not imply the semifilter trichotomy principle by constructing a specific forcing extension where the trichotomy fails, clarifying their logical relationship.

Contribution

It proves that the inequality 67 < 68 is strictly stronger than the filter dichotomy principle, providing a counterexample in a forcing extension.

Findings

A forcing extension where every non-meagre filter is ultra by finite-to-one.

The semifilter trichotomy does not hold in this extension.

The inequality 67 < 68 is strictly stronger than the filter dichotomy.

Abstract

We answer Blass' question from 1989 of whether the inequality $\gu < \gro$ is strictly stronger than the filter dichotomy principle affirmatively. We show that there is a forcing extension in which every non-meagre filter on $\omega$ is ultra by finite-to-one and the semifilter trichotomy does not hold. This trichotomy says: every semifilter is either meagre or comeagre or ultra by finite-to-one. The trichotomy is equivalent to the inequality $\gu<\gro$ by work of Blass and Laflamme. Combinatorics of block sequences is used to establish forcing notions that preserve suitable properties of block sequences.