For any sentence φ that characterizes a structure M up to isomorphism, the class of Henkin models Mod_H(φ) contains models other than M itself in all non-trivial cases

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, infini