More Derived Models in PFA

TL;DR

This work investigates how strong forcing axioms, particularly $PFA$, constrain inner models satisfying $AD^+$ via derived models at a limit of Woodin cardinals. It develops Covering Matrix techniques and the related $CP(\kappa^+,\lambda)$ property to bound $\Theta^{D(V,\kappa)}$, and establishes $D(V,\kappa)\models AD_R$ under certain large-cardinal and mouse-capturing assumptions; it also proves a detailed Wilson-style result for weakly compact limits of Woodin cardinals. The paper then provides partial progress towards Woodin's conjecture by proving $AD_R$ under $PFA$ under a hod-pair/mouse-capturing framework, and shows that if $\kappa$ is indestructibly weakly compact, the derived model already satisfies $AD_R$. Together, these results deepen the connections between forcing axioms, large cardinals, and determinacy in derived inner models, and highlight both the power and the current limitations of these methods.

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 $κ$. In particular, using a concept called Covering Matrices, we show that the $Θ$ of the derived model at $κ$ is strictly less than $κ^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $κ$ is a limit of Woodin cardinals of cofinality $ω$ 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 $κ$ is a regular limit of Woodin cardinals. If $κ$ is an indestructibly $(κ,κ^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$.

Theorems & Definitions (53)

• Conjecture 1.1: PFA
• Theorem 2.1: Wilson
• Definition 2.2: Viale
• Definition 2.3: Viale
• Theorem 2.4
• proof
• Claim 2.5
• proof
• Claim 2.6
• proof