Let X be a Polish space, and let ${\left(A_p\right)}_{p\in \omega }$ be a sequence of $G_{\delta }$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether ${\cup }_{p\in \omega }{A}_p$ is a true ${\sum }_3^0$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true ${\sum }_3^0$.