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.

arXiv comment21 pages. This version: March 2019