An incompleteness theorem via ordinal analysis

We present an analogue of Gödel’s second incompleteness theorem. Whereas Gödel showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.