  There is no infinite sequence of $\Pi^1_1$-sound extensions of
$\mathsf{ACA}_0$ each of which proves $\Pi^1_1$-reflection of the next. This
engenders a well-founded ``reflection ranking'' of $\Pi^1_1$-sound extensions
of $\mathsf{ACA}_0$. For any $\Pi^1_1$-sound theory $T$ extending
$\mathsf{ACA}^+_0$, the reflection rank of $T$ equals the proof-theoretic
ordinal of $T$. This provides an alternative characterization of the notion of
``proof-theoretic ordinal,'' which is one of the central concepts of proof
theory. In this note we provide an alternative proof of this theorem using
cut-elimination for infinitary derivations.
