In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).
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Existence
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.
Importance
Non-standard models are studied in set theory, non-standard analysis and non-standard models of arithmetic.
References
Non-standard model Wikipedia(Text) CC BY-SA