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TREES AND STATIONARY REFLECTION AT DOUBLE SUCCESSORS OF REGULAR CARDINALS

Part of: Set theory

Published online by Cambridge University Press:  14 February 2022

THOMAS GILTON ,
MAXWELL LEVINE  and
ŠÁRKA STEJSKALOVÁ Open the ORCID record for ŠÁRKA STEJSKALOVÁ [Opens in a new window]
THOMAS GILTON*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF PITTSBURGH 166 THACKERAY AVENUE PITTSBURGH, PA 15213, USA
MAXWELL LEVINE
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF FREIBURG ERNST-ZERMELO-STRASSE 1 FREIBURG IM BREISGAU 79104, GERMANY E-mail: maxwell.levine@mathematik.uni-freiburg.de
ŠÁRKA STEJSKALOVÁ
Affiliation:
DEPARTMENT OF LOGIC CHARLES UNIVERSITY CELETNÁ 20, 116 42 PRAGUE 1, CZECH REPUBLIC E-mail: sarka.stejskalova@ff.cuni.cz
*
E-mail: tdg25@pitt.edu
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Abstract

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa $ , updating some classical constructions in the process. This includes models of $\mathsf {CSR}(\kappa ^{++})\wedge {\sf TP}(\kappa ^{++})$ (both with and without ${\sf AP}(\kappa ^{++})$ ) and models of the conjunctions ${\sf SR}(\kappa ^{++}) \wedge \mathsf {wTP}(\kappa ^{++}) \wedge {\sf AP}(\kappa ^{++})$ and $\neg {\sf AP}(\kappa ^{++}) \wedge {\sf SR}(\kappa ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author [9], and is here given using different methods). Analogs of these results with the failure of $\sf {SH}(\kappa ^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^\kappa $ , applying recent joint work by Honzik and the third author.

Keywords

forcingtree propertystationary reflectionclub stationary reflectionsuslin treesapproachability

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 03E35: Consistency and independence results 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 2 , June 2023 , pp. 780 - 810
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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