{wiki=Overspill}

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.

Overspill lemma

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

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 120 / 2 / b / Solution

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.


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 Incoming links: Standard cut of a nonstandard model of arithmetic