Formal systems have inherent weakness, traceable to Skolem and Gödel, that prevents deductive systems from pinning down infinite structures up to isomorphism

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