Published online by Cambridge University Press: 08 January 2021
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 on ordinals. We show that this order of linear reflection is a prewellordering of length
$\omega ^{\omega }$. This requires considering the relationship between linear and some non-linear reflection patterns, such as
$\sigma \wedge \pi $, the pattern of simultaneous
$\Sigma ^{1}_{1}$- and
$\Pi ^{1}_{1}$-reflection. The proofs involve linking the lengths of
$\alpha $-recursive wellorderings to various forms of stability and reflection properties satisfied by ordinals
$\alpha $ within standard and non-standard models of set theory.