Non-standard model

In model theory, a nonstandard model is a model that has a different (larger) underlying set than the "standard" model. The sentences that were true in the standard model must be true in the nonstandard model (by the definition of its being a model), however, in the new model, new sentences will arise from the new elements in the larger underlying set. In model theoretic terms, a nonstandard model is an elementary extension of a standard model.

Examples of nonstandard models are those of non-standard analysis, and non-standard arithmetic.