Free sequences in P(ω)/fin

Abstract

We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(\omega )/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of Monk, we demonstrate the consistency of ${\omega }_1=\mathfrak{i}=\mathfrak{f}<\mathfrak{u}={\omega }_2$. In fact, this consistency is demonstrated in the model of S. Shelah for $\mathfrak{i}<\mathfrak{u}$. Our paper provides a streamlined and mostly self contained presentation of this construction.