Derived Models in PFA

TL;DR

This paper investigates a conjecture by Wilson regarding the size of the derived model at a cardinal under certain axioms, proving it in specific cases and under additional assumptions.

Contribution

It proves Wilson's conjecture for the old derived model and extends the result to the new derived model assuming mouse capturing, providing new insights into the structure of derived models.

Findings

The conjecture holds for the old derived model.

Under mouse capturing, the conjecture holds for the new derived model.

In the old derived model, ta < ta^+.

Abstract

We discuss a conjecture of Wilson that under the proper forcing axiom, $\Theta_0$ of the derived model at $\kappa$ is below $\kappa^+$. 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 $\Theta < \kappa^+$ in the case of the old derived model, and under additional hypotheses for the new derived model.