Journal of Symbolic Logic 83 (1):349-371 (2018)

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 aboveκ.If in additionκis supercompact then we can forceκto 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${\aleph _\omega }$.
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DOI 10.1017/jsl.2017.69
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References found in this work BETA

The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Some Exact Equiconsistency Results in Set Theory.Leo Harrington & Saharon Shelah - 1985 - Notre Dame Journal of Formal Logic 26 (2):178-188.
Reflecting Stationary Sets and Successors of Singular Cardinals.Saharon Shelah - 1991 - Archive for Mathematical Logic 31 (1):25-53.

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