Forcing Diamond and Applications to Iterability

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 $\kappa ={\kappa }^{<\kappa }\ge {\aleph }_1$ 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.