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

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.