# THE EIGHTFOLD WAY

@article{Cummings2018THEEW,
title={THE EIGHTFOLD WAY},
author={James Cummings and Sy-David Friedman and Menachem Magidor and Assaf Rinot},
journal={The Journal of Symbolic Logic},
year={2018},
volume={83},
pages={349 - 371}
}
• Published 1 March 2018
• Mathematics
• The Journal of Symbolic Logic
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…
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