The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova.

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 $\kappa$.
If in addition $\kappa$ is supercompact then we can force $\kappa$ to be ${\mathrm{\aleph }}_{\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a non-reflecting 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 ${\mathrm{\aleph }}_{\omega }$.

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J. Cummings, S.-D. Friedman, M. Magidor, A. Rinot and D. Sinapova, The eightfold way, J. Symbolic Logic, 83(1): 349-371, 2018.