THE ORDER OF REFLECTION

Extending Aanderaa's classical result that $\pi^1_1<\sigma^1_1$, we determine the order between any two patterns of iterated $\Sigma^1_1$- and $\Pi^1_1$-reflection. We show this \emph{linear reflection order} is a prewellordering length $\omega^\omega$. This requires considering relationship linear some \emph{non-linear} patterns, such as $\sigma^1_1\wedge\pi^1_1$, pattern simultaneous