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THE ORDER OF REFLECTION

Published online by Cambridge University Press:  08 January 2021

JUAN P. AGUILERA*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF GHENT KRIJGSLAAN 281-S8, 9000 GHENT, BELGIUM and INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRASSE 8–10, 1040 VIENNA, AUSTRIAE-mail: [email protected]

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.

Type
Article
Copyright
© Association for Symbolic Logic 2021

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