 Model-theoretic interpretations are sets: a model is an ordered pair of a non-empty set and an interpretation function. But since modern  set theory  proves that there is no universal set, no model can ever interpret the quantifiers by means of a universal domain of discourse. It may seem, then, that there are interpretations of the language of pure quantificational logic to which no model corresponds. And this raises the question of whether truth in all models may fall short of truth under all interpretations of the language.