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

First establish the transfer of provability requested in the hint. The propositional theorem $X\to (\neg G\to X)$ and the derivability conditions giveThe supplied arithmetized deduction equivalence identifies the right-hand provability assertion, within $S^{\ast }$, with ${\square }^{\ast }X$. HenceChoose a fixed contradiction $\perp$ for $X$. Part (b) gives $S⊢G\leftrightarrow \neg \square \perp$, so adjoining $\neg G$ gives $S^{\ast }⊢\square \perp$. The transfer implication now gives $S^{\ast }⊢{\square }^{\ast }\perp$.

Apply (b) inside $S^{\ast }$ to its own Gödel sentence: it proves $G^{\ast }\leftrightarrow \neg {\square }^{\ast }\perp$. The last provability assertion therefore entailsThus the extension proves the negation of its own Gödel sentence as well. When $S$ is first-order consistent, (c) says this phenomenon occurs in a first-order consistent extension; the extension asserts an internal formal proof of contradiction without there being an actual standard formal proof of contradiction.