 Obviously, \(\Mod(\phi)\subseteq \textit{Mod}_H(\phi)\). If \(\phi\) characterizes \(\mm\) up to isomorphism, then \(\mm\in \Mod(\phi) \). In all non-trivial  cases[16]  \(\textit{Mod}_H(\phi)\ne \{\mm\}\). We can think of \(\textit{Mod}_H(\phi)\) as a class of “approximations” of \(\mm\). One of the approximations is the “real” \(\mm\) but by means of deductions in second-order logic we cannot tell which. Because of the inherent weakness of formal systems, going back to Skolem and Gödel, infinite structures are shrouded by Henkin models and cannot be gotten perfectly into focus by deductive means.