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arXiv · 2504.08390

Forcing Diamond and Applications to Iterability

Heike Mildenberger·Saharon Shelah
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Abstract

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $\kappa = \kappa^{<\kappa} \geq\aleph_1$ implies that $\kappa$-supported iterations of $\kappa$-Sacks forcing do not collapse $\kappa^+$ and are $\kappa$-proper in the affirmative. The results pertain to other higher tree forcings.

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Heike Mildenberger, Saharon Shelah. 2025-04-11. Forcing Diamond and Applications to Iterability. https://arxiv.org/abs/2504.08390

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