  Ordinal analysis induces a partition of $\Sigma^1_1$-definable and
$\Pi^1_1$-sound theories whereby two theories are equivalent if they have the
same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is
finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$
and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$
of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a
single distinction that the ordinal analysis partition does not make.

Logic