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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.

No such formula exists. If $\varphi (x)$ defined precisely the standard cut of a nonstandard model of arithmetic, then $\mathcal{M}\models \varphi (n)$ for every standard natural number $n$. The overspill lemma would produce a nonstandard $b\in M$ satisfying $\varphi (b)$, contradicting the proposed definition. Thus the standard elements form an external, nondefinable subset of every nonstandard model of Peano arithmetic.