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

For unrestricted consistency self-strengthening is inconsistent, again take $A=S+\neg {\mathrm{Con}}_S$. In $S^{++}$ the axiom ${\mathrm{Con}}_S$ and the preceding implication prove ${\mathrm{Con}}_A$. The additional rule then proves ${\mathrm{Con}}_{A+{\mathrm{Con}}_A}$.

However consistency of an extension implies consistency of its subtheory by the formalized inclusion of proofs: $S⊢{\mathrm{Con}}_A\to {\mathrm{Con}}_S$. Thus the ordinary theory $A+{\mathrm{Con}}_A$ contains $\neg {\mathrm{Con}}_S$ and derives ${\mathrm{Con}}_S$, so it has a finite contradiction proof. Checking that concrete proof in arithmetic gives $S⊢\neg {\mathrm{Con}}_{A+{\mathrm{Con}}_A}$. This theorem is also present in $S^{++}$, contradicting the result of the new rule. Therefore $S^{++}$ is inconsistent.

The error in the added rule is its unrestricted internal use on arbitrary arithmetically presented systems, including systems containing false consistency assertions. A true external consistency progression for a specified sound system is not this unrestricted inference rule.