We begin this talk by showing how, through work of Marde\v{s}i\'{c} and Prasolov, the study of the additivity of strong homology naturally leads to questions about the vanishing of the derived limits of a particular inverse system of abelian groups indexed by ${^\omega}\omega$, classically denoted by $\mathbf{A}$. We then survey a number of results indicating the sensitivity of $\lim^1 \mathbf{A}$ to the axioms of set theory. In particular, we will present and give proofs of a result of Dow, Simon, and Vaughan stating that, if $\mathfrak{d} = \aleph_1$, then $\lim^1 \mathbf{A} \neq 0$, and a complementary result of Todorcevic stating that the Open Coloring Axiom implies that $\lim^1 \mathbf{A} = 0$.