Lemma 2.2.15 (Diagonalisation Lemma). Assuming that:

• $T$ an $L_{\mathrm{PA}}$-theory

• in $T$, every total recursive function is ${\mathrm{\Sigma }}_1$-represented

• $𝜃(x)$ an $L_{\mathrm{PA}}$-formula with one free variable $x$