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Overspill is the principle that an internal subset of a nonstandard extension containing every standard natural number must contain some nonstandard natural number. The overspill lemma is its definable-set instance in nonstandard arithmetic.

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.