We continue to explore the sensitivity of the additivity of strong homology to set-theoretic assumptions. This sensitivity centers on the behavior of the higher derived limits of an inverse system A. We'll review definitions of this system and those limits, and discuss two main results: 1. Under PFA, strong homology is not additive; 2. $\lim^1 A=0$ if and only if $\lim^1 X=0$ for all $X$ in a much wider class of inverse systems.