Successive failures of approachability

Abstract

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.