  It is well-known that natural axiomatic theories are pre-well-ordered by
logical strength, according to various characterizations of logical strength
such as consistency strength and inclusion of $\Pi^0_1$ theorems. Though these
notions of logical strength coincide for natural theories, they are not
generally equivalent. We study analogues of these notions -- such as
$\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the
presence of an oracle for $\Sigma^1_1$ truths. In this context these notions
coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and
may drop the non-mathematical quantification over "natural" theories.
