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THE EIGHTFOLD WAY

Published online by Cambridge University Press:  01 May 2018

JAMES CUMMINGS ,
SY-DAVID FRIEDMAN ,
MENACHEM MAGIDOR ,
ASSAF RINOT  and
DIMA SINAPOVA
JAMES CUMMINGS
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA15213-3890, USA E-mail:jcumming@andrew.cmu.eduURL: http://www.math.cmu.edu/users/jcumming/
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNAA-1090VIENNA, AUSTRIA E-mail:sdf@logic.univie.ac.atURL: http://www.logic.univie.ac.at/∼sdf/
MENACHEM MAGIDOR
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM91904, ISRAEL E-mail:mensara@savion.huji.ac.il
ASSAF RINOT
Affiliation:
DEPARTMENT OF MATHEMATICS BAR-ILAN UNIVERSITY RAMAT-GAN 5290002, ISRAEL E-mail:rinotas@math.biu.ac.ilURL: http://www.assafrinot.com
DIMA SINAPOVA
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO CHICAGO, IL 60607-7045, USA E-mail: sinapova@math.uic.eduURL: http://homepages.math.uic.edu/∼sinapova
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Abstract

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at ${\kappa ^{ + + }}$, assuming that $\kappa = {\kappa ^{ < \kappa }}$ and there is a weakly compact cardinal above κ.

If in addition κ is supercompact then we can force κ to be ${\aleph _\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$.

Keywords

Primary 03E35Secondary 03E55squareAronszajn treetree propertystationary reflectionapproachabilityMitchell forcingPrikry forcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 349 - 371
Copyright
Copyright © The Association for Symbolic Logic 2018 

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