Past exam of the Gödel's incompleteness theorems course of the University of Oxford / 2011 / 2 / c

Use just the assumed local reflection instance with $Y=\neg X$:By the deduction theorem for propositional logic, $S⊢X\to (B_S(\neg X)\to \neg X)$. Classical propositional logic makes this imply the unconditional reflection statement $S⊢B_S(\neg X)\to \neg X$: if $X$ is false the consequent is already true, and if $X$ is true the displayed implication supplies it. Löb's theorem applied to $\neg X$ now givesThus full local reflection in a one-sentence extension is inconsistent; in this argument even the single instance concerning $\neg X$ suffices.