More Derived Models in PFA

Abstract

This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\hat{I}º$. In particular, using a concept called Covering Matrices, we show that the $\hat{I}˜$ of the derived model at $\hat{I}º$ is strictly less than $\hat{I}{º}^{+}$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $\hat{I}º$ is a limit of Woodin cardinals of cofinality $\ddot{I}‰$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $A{D}_{\mathbb{R}}$ under $PFA$ when $\hat{I}º$ is a regular limit of Woodin cardinals. If $\hat{I}º$ is an indestructibly $(\hat{I}º,\hat{I}{º}^{+})$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $A{D}_{\mathbb{R}}$.