A nonstandard model of Peano arithmetic is a model not isomorphic to the standard natural-number structure. It contains the standard numerals as an initial segment and also contains elements larger than every standard numeral.
In a model of Peano arithmetic, an element realizes this partial type exactly when it is not a standard numeral. A model omitting it is isomorphic to the standard natural-number structure. For a complete consistent extension of arithmetic, the type is nonprincipal exactly when is the theory of true arithmetic: apply the omitting types theorem in one direction and use an actual standard witness to refute any proposed isolating formula in the other. For incomplete theories, a principal partial type can still be omitted by some models.
The standard cut is the external initial segment consisting of the interpretations of the standard numerals. It is not definable, even with parameters, inside the nonstandard model.
If a definable property in a nonstandard model of Peano arithmetic holds of every standard natural number, then it holds of some nonstandard element. Equivalently, a definable set containing the entire standard cut of a nonstandard model of arithmetic must overspill beyond it.
This is the arithmetic case of overspill, which also applies to internal subsets of nonstandard extensions in analysis.
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Non-standard models of arithmetic are structures that satisfy the axioms of Peano arithmetic (PA) but contain "non-standard" elements that do not correspond to the standard natural numbers (0, 1, 2, ...). In other words, while a standard model of arithmetic consists only of the usual natural numbers, a non-standard model includes additional "infinitely large" and "infinitesimally small" numbers that do not have a counterpart in the standard model.