More Derived Models in PFA

TL;DR

This paper advances understanding of the derived model at Woodin cardinals, showing under certain axioms and conditions that the model satisfies determinacy hypotheses and analyzing the size of the model's $ heta$ using Covering Matrices.

Contribution

It introduces Covering Matrices to analyze the derived model at Woodin cardinals and establishes conditions under which the model satisfies $AD_{\mathbb{R}}$ and bounds on $ heta$.

Findings

The $ heta$ of the derived model is less than $oldsymbol{ ext{cardinal}}^+$ under various circumstances.

Under $PFA$, the derived model satisfies $AD_{\mathbb{R}}$ when $oldsymbol{ ext{limit of Woodin cardinals}}$ has cofinality $oldsymbol{ ext{omega}}$.

If $oldsymbol{ ext{indestructibly weakly compact limit of Woodin cardinals}}$, then the derived model satisfies $AD_{\mathbb{R}}$.

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 $\kappa$. In particular, using a concept called Covering Matrices, we show that the $\Theta$ of the derived model at $\kappa$ is strictly less than $\kappa^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $\kappa$ is a limit of Woodin cardinals of cofinality $\omega$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $AD_{\mathbb{R}}$ under $PFA$ when $\kappa$ is a regular limit of Woodin cardinals. If $\kappa$ is an indestructibly $(\kappa,\kappa^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$.