The eightfold way

James Cummings; Sy David Friedman; Menachem Magidor; Assaf Rinot; Dima Sinapova

doi:10.1017/jsl.2017.69

The eightfold way

James Cummings, Sy David Friedman, Menachem Magidor, Assaf Rinot, Dima Sinapova

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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 κ⁺⁺ assuming that κ = κ ^<κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be N_ωin the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a κ⁺⁺-Souslin tree, variants of Mitchell's forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into N_ω.

Original language	English
Pages (from-to)	349-371
Number of pages	23
Journal	Journal of Symbolic Logic
Volume	83
Issue number	1
DOIs	https://doi.org/10.1017/jsl.2017.69
State	Published - 1 Mar 2018

Bibliographical note

Publisher Copyright:
© 2018 The Association for Symbolic Logic.

Funding

Cummings was partially supported by the National Science Foundation, grants DMS-1101156 and DMS-1500790. Friedman would like to thank the Austrian Science Fund for its generous support through the research project P25748. Rinot was partially supported by the Israel Science Foundation, grant 1630/14. Sinapova was partially supported by the National Science Foundation, grants DMS-1362485 and Career-1454945. The results in this article were conceived and proved during three visits to the American Institute of Mathematics, as part of the Institute's SQuaRE collaborative research program. The authors are deeply grateful to Brian Conrey, Estelle Basor and all the staff at AIM for providing an ideal working environment.

Funders	Funder number
American Institute of Mathematics
Austrian Science Fund	P25748
Israel Science Foundation	1630/14
National Science Foundation	DMS-1362485, Career-1454945, DMS-1101156, DMS-1500790

Keywords

Aronszajn tree
Mitchell forcing
Prikry forcing
approachability
square
stationary reflection
tree property

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10.1017/jsl.2017.69

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AB - 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 κ++ assuming that κ = κ <κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be Nωin the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a κ++-Souslin tree, variants of Mitchell's forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into Nω.

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The eightfold way

Abstract

Bibliographical note

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Keywords

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