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THE ORDER OF REFLECTION

Part of: Computability and recursion theory Set theory

Published online by Cambridge University Press:  08 January 2021

JUAN P. AGUILERA
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: aguilera@logic.at
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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.

Keywords

reflecting ordinalstabilityα-recursionGandy ordinalpattern of reflection

MSC classification

Primary: 03D60: Computability and recursion theory on ordinals, admissible sets, etc. 03E45: Inner models, including constructibility, ordinal definability, and core models
Secondary: 03D40: Word problems, etc. 03E10: Ordinal and cardinal numbers
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1555 - 1583
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Copyright
© Association for Symbolic Logic 2021

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