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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.

Richard Kaye works on mathematical logic and arithmetic model theory. His exposition of Tennenbaum's theorem studies why countable nonstandard arithmetic cannot have an ordinary computable presentation with computable operations.^[ref]

The proof encodes a standard nonrecursive set inside a nonstandard model, using recursively inseparable sets and overspill. If the model's arithmetic were computable in the required way, the coded set could be recovered algorithmically, giving a contradiction. Kaye's account makes the interaction between model theory and computability theory explicit.^[ref]