  We present an analogue of G\"{o}del's second incompleteness theorem for
systems of second-order arithmetic. Whereas G\"{o}del 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.
