 If \(\phi\) characterizes a model \(\ma\), then \(\Mod(\phi)\) is just the class of models isomorphic to \(\ma\). If we look at \(\Mod(\phi)\) from the perspective of set theory, we know it is \(\Delta_2\)  (Theorem 4).  On the other hand, the set of Gödel numbers of valid second-order sentences is \(\Pi_2\)-complete  (Theorem 4),  hence not \(\Sigma_2\) and in particular, not \(\Delta_2\). This contemplation leads us to the following extension of  Theorem 3  above: