Non-standard model

In model theory, a nonstandard model is a model of a theory that is not isomorphic to the intended model (or standard model). If the intended model is infinite and the language is first-order, then the Löwenheim-Skolem theorems guarantee the existence of nonstandard models. The nonstandard models can be chosen as elementary extensions or elementary substructures of the intended model.

Nonstandard models are studied in set theory, non-standard analysis, and non-standard arithmetic.