An ω-model of "T" is a model of "T" whose domain includes the natural numbers and whose specified names and symbol "N" are standardly interpreted, respectively as those numbers and the predicate having just those numbers as its domain (whence there are no nonstandard numbers). If "N" is absent from the language then what would have been the domain of "N" is required to be that of the model, i.e. the model contains only the natural numbers. (Other models of "T" may interpret these symbols nonstandardly; the domain of "N" need not even be countable, for example.) These requirements make the ω-rule sound in every ω-model. As a corollary to the omitting types theorem, the converse also holds: the theory "T" has an ω-model if and only if it is consistent in ω-logic.