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arXiv · 1910.05159

Perfect Subtree Property for Weakly Compact Cardinals

Yair Hayut·Sandra Müller
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Abstract

We investigate the consistency strength of the statement: $κ$ is weakly compact and there is no tree on $κ$ with exactly $κ^{+}$ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly $κ^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal $κ$ the nonexistence of a tree on $κ$ with exactly $κ^{+}$ many branches and, in particular, the Perfect Subtree Property for $κ$, implies the consistency of $AD_{\mathbb{R}} + DC$.

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Yair Hayut, Sandra Müller. 2021-09-21. Perfect Subtree Property for Weakly Compact Cardinals. https://arxiv.org/abs/1910.05159

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