  In the study of strong homology Marde\v{s}i\'c and Prasolov isolated a
certain inverse system of abelian groups $\mathbf A$ indexed by elements of
$\omega^\omega$. They showed that if strong homology is additive on a class of
spaces containing closed subsets of Euclidean spaces then the higher derived
limits $\lim^n \mathbf A$ must vanish, for $n>0$. They also proved that under
the Continuum Hypothesis $\lim^1 \mathbf A \neq 0$. The question whether
$\lim^n \mathbf A$ vanishes, for $n>0$, has attracted considerable interest
from set theorists. Dow, Simon and Vaughan showed that under PFA $\lim^1
\mathbf A =0$. Bergfalk show that it is consistent that $\lim^2\mathbf A$ does
not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly
compact cardinal, it is relatively consistent with ZFC that $\lim^n \mathbf A
=0$, for all $n$. The large cardinal assumption was recently removed by
Bergfalk, Hru\v{s}ak and Lambie-Henson. We complete the picture by showing
that, for any $n>0$, it is relatively consistent with ZFC that $\lim^n \mathbf
A \neq 0$.

Logic;Category Theory