Past exam of the Gödel's incompleteness theorems course of the University of Oxford / 2011 / 1 / e

Let $T=S+\{{\mathrm{Con}}_S\}$ and assume $T$ is consistent. The allowed arithmetizations of the previous equivalences giveSuppose $S⊢R\to G$. Its contrapositive is $\neg G\to \neg R$, and the Hilbert-Bernays-Löb derivability conditions then giveCombining with the two internal equivalences yields $S⊢{\mathrm{Con}}_S\to \mathrm{Con}(T)$. Since $T$ contains ${\mathrm{Con}}_S$, it would prove $\mathrm{Con}(T)$, contrary to the given Gödel second incompleteness theorem for this consistent extension. HenceTogether with the preceding implication, this is the Strict Gödel-to-Rosser implication under the stated consistency hypothesis.