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

 Solution (source code)