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High Quality Content by WIKIPEDIA articles! In mathematical logic, a nonstandard model of arithmetic is a model of (first-order) Peano arithmetic that contains nonstandard numbers. The standard model of arithmetic consists of the set of standard natural numbers {0, 1, 2, }. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A nonstandard model is one that has additional elements outside this initial segment. The existence of such models is due to Thoralf Skolem (1934).The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem. To do this, a set of axioms P is defined in a language including the language of Peano arithmetic together with a new constant symbol x.