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