  Motivated by showing that in ZFC we cannot construct a special Aronszajn tree
on some cardinal greater than $\aleph_1$, we produce a model in which the
approachability property fails (hence there are no special Aronszajn trees) at
all regular cardinals in the interval $[\aleph_2, \aleph_{\omega^2+3}]$ and
$\aleph_{\omega^2}$ is strong limit.
