Forcing Diamond and Applications to Iterability
Heike Mildenberger, Saharon Shelah
TL;DR
This work investigates diamond-adding phenomena for higher tree forcings at uncountable cardinals. By analyzing higher Sacks, Silver, Miller, and Laver forcings and their $W$-variants, it proves that diamonds can be forced and that $\leq\kappa$-supported iterations preserve $\kappa^+$ and are $\kappa$-proper under the hypothesis $\kappa^{<\kappa}=\kappa\geq\aleph_1$, with different methods for weakly Mahlo and regular limit $\kappa$. It provides explicit diamond-name constructions, including simple forms for Miller/Laver, and develops a strategic-closure framework to derive diamonds under strong closure hypotheses. The paper also explains how approachability and Bernstein techniques yield diamonds and collapses in broader contexts, including $W$-variants and successor cardinals, thereby extending iterability and preservation results for a wide class of higher tree forcings. Overall, it advances our understanding of diamond principles in forcing at $\kappa$ and supplies robust methods for proving $\kappa$-properness and non-collapse in high-cardinal iterations.
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 $κ= κ^{<κ} \geq\aleph_1$ implies that $κ$-supported iterations of $κ$-Sacks forcing do not collapse $κ^+$ and are $κ$-proper in the affirmative. The results pertain to other higher tree forcings.