We give some background to a number of independence results surrounding the question of the additivity of strong homology. These results center on the question of the vanishing of the higher derived limits of an inverse system $\mathbb{A}$ indexed by the functions from $\omega$ to $\omega$. Time permitting, we'll show that $\text{lim}^1\mathbb{A}=0$ if and only if $\text{lim}^1\mathbb{A}_\kappa=0$, where $\mathbb{A}_\kappa$ is $\mathbb{A}$'s generalization to $\omega^\kappa$, with $\kappa>\omega$ arbitrary.