Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / A / 1 / d / Solution

Put $T=S\cup \{\neg G_S\}$. The first supplied equivalence, retained in $T$, combines with its new axiom to giveThe supplied implication $\mathrm{Con}(T)\to \mathrm{Con}(S)$ then yields, by contraposition,Finally the second supplied equivalence is $T⊢G_T\leftrightarrow \mathrm{Con}(T)$, so propositional reasoning givesNo extra first-order consistency assumption is needed for this formal derivation. If $S$ is first-order consistent, part (b) shows that adding $\neg G_S$ preserves first-order consistency, because inconsistency of the extension would imply $S⊢G_S$. Such a first-order consistent extension therefore proves the negation of its own true Gödel sentence, illustrating why first-order consistency alone does not imply one-consistency.