Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / A / 3 / b / Solution

Abbreviate $P(┌A┐)$ by $\square A$. Since $S⊢\neg X$, it proves $X\to G$. The theoremhood and distribution conditions giveThe fixed-point equation gives $S⊢G\to (\square G\to X)$, because $G$ implies $\neg \square G$ and a false antecedent implies any conclusion. Applying theoremhood and distribution yieldsPositive introspection gives $\square G\to \square \square G$; distribution then gives $\square G\to \square X$. Thus $S⊢\square G\leftrightarrow \square X$. Combining with $G\leftrightarrow \neg \square G$ provesIn particular, choosing a fixed logical contradiction for $X$ identifies this Gödel sentence with the formal consistency statement of $S$.