Derived Models in PFA

Abstract

We discuss a conjecture of Wilson that under the proper forcing axiom, $\hat{I}{˜}_0$ of the derived model at $\hat{I}º$ is below $\hat{I}{º}^{+}$. We prove the conjecture holds for the old derived model. Assuming mouse capturing in the new derived model, the conjecture holds there as well. We also show $\hat{I}˜<\hat{I}{º}^{+}$ in the case of the old derived model, and under additional hypotheses for the new derived model.