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

The diagonal lemma says that for every one-variable formula $\theta (x)$ there is a sentence $\gamma$ such thatLet $d(n)$ be the computable function taking the code of a one-variable formula $\alpha (x)$ to the code of $\alpha (\bar{n})$. By the assumed representation theorem, choose a Sigma-1 formula $D(x,y)$ representing $d$. Given $\theta$, putand let $b=┌\beta ┐$. Taking $\gamma =\beta (\bar{b})$, representability proves in $T$ that the unique relevant $y$ is $d(b)=┌\gamma ┐$, yielding the required equivalence.