 It is illuminating to reflect on the first incompleteness theorem also from the model theoretic perspective—though the theorem itself does not in any way require this. Namely, it is possible to conclude that any theory \(F\) satisfying the conditions of the theorem must possess, in addition to the intended interpretation or “standard model” (in the case of arithmetical theories, the structure of natural numbers), non-intended interpretations or “non-standard models”—that no such theory can rule out the latter and fix uniquely the intended interpretation. Namely, if there are independent statements such as \(G_F, F\) must have both models which satisfy \(G_F\) and models which rather satisfy \(\neg G_F\). As \(\neg G_F\) is equivalent to  \(\exists x\Prf_F (x, \ulcorner G_F\urcorner)\), the latter models must possess entities which satisfy the formula \(\Prf_F (x, \ulcorner G_F\urcorner)\). And yet we know (because \(\Prf_F (x, y)\) strongly represents the proof relation) that for any numeral \(\underline{n}, F\) can prove \(\neg\Prf_F (\underline{n}, \ulcorner G_F\urcorner)\). Therefore, no natural number \(\boldsymbol{n}\) can witness the formula. It follows that any such non-standard model must contain, in addition to natural numbers (denotations of the numerals \(\underline{n})\), “infinite” non-natural numbers after the natural numbers.