The Order of Reflection

Abstract

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 that this \emph{linear reflection order} is a prewellordering of length $\omega^\omega$. This requires considering the relationship between linear and some \emph{non-linear} reflection patterns, such as $\sigma^1_1\wedge\pi^1_1$, the pattern of simultaneous $\Sigma^1_1$- and $\Pi^1_1$-reflection.