Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / A / 1 / a

Write $\square A$ for the arithmetical sentence asserting $S$-provability of $A$. If $S⊢G$, the first derivability condition gives $S⊢\square G$. The stipulated fixed-point equivalence also gives $S⊢\neg \square G$. This contradicts consistency. Conversely an inconsistent classical theory proves every sentence, including $G$. ThusOnly the direction from an actual proof to its formalized provability is used here. There is no inference from a theorem $\square G$ to a theorem $G$, and no assumption that every theorem of $S$ is true.