= Solution
Suppose such a formula $\theta(x)$ existed. Apply the <Diagonal lemma> to $\neg\theta(x)$ to obtain a sentence $\sigma$ for which
$$
\mathrm{PA}^-\vdash\sigma\leftrightarrow\neg\theta(\ulcorner\sigma\urcorner).
$$
Because $M\models\mathrm{PA}^-$, the equivalence holds in $M$. But the defining property of $\theta$ says $M\models\theta(\ulcorner\sigma\urcorner)$ exactly when $M\models\sigma$, producing $M\models\sigma$ exactly when $M\not\models\sigma$, a contradiction.
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