Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 120 / 3 / d / Solution

Suppose such a formula $\theta (x)$ existed. Apply the Diagonal lemma to $\neg \theta (x)$ to obtain a sentence $\sigma$ for whichBecause $M\models {\mathrm{PA}}^{-}$, the equivalence holds in $M$. But the defining property of $\theta$ says $M\models \theta (┌\sigma ┐)$ exactly when $M\models \sigma$, producing $M\models \sigma$ exactly when $M/\models \sigma$, a contradiction.