If the intended model is infinite and the language is first-order, then the Löwenheim–Skolem theorems guarantee the existence of non-standard models. The non-standard models can be chosen as elementary extensions or elementary substructures of the intended model.
Non-standard models are studied in set theory, non-standard analysis and non-standard models of arithmetic.
Roman Kossak, 2004 Nonstandard Models of Arithmetic and Set Theory American Mathematical Soc. ↩