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THE ORDER OF REFLECTION
Published online by Cambridge University Press: 08 January 2021
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 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.
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- © Association for Symbolic Logic 2021
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