The Largest Suslin Axiom

TL;DR

The work develops a comprehensive fine-structure theory for the minimal model of the Largest Suslin Axiom ($\sf{LSA}$), leveraging a rich framework of Hybrid ${\mathcal J}$-structures and hod-like premice. It establishes that the minimal model satisfies the Mouse Set Conjecture and demonstrates the existence of the minimal model under $\sf{PFA}$ and certain large-cardinal hypotheses, tying determinacy, inner model theory, and forcing axioms together. A key methodological innovation is the introduction of short-tree strategy mice (STS mice) and the un-dropping machinery, which together enable precise control of branches and robustness of comparison arguments in the $\sf{AD}^+$ setting. These advances create a unified perspective on hod-like structures, Solovay hierarchy levels, and the layering of premice, with potential implications for core model induction beyond $\sf{LSA}$ and the interaction between determinacy and large-cardinal theories.

Abstract

We develop the basic fine structure theory of the minimal model of the Largest Suslin Axiom. In particular, we prove that that the minimal model of the Largest Suslin Axiom satisfies the Mouse Set Conjecture, and that the Proper Forcing Axiom implies the minimal model of the Largest Suslin Axiom exists.

Figures (10)

• Figure 2.7.1: Lsa type $\sf{lses}$. Here, ${\mathcal{P} }$ is an lsa type $\sf{lses}$. $\kappa$ is a limit of Woodin cardinals in ${\mathcal{P} }$, $\delta=\delta^{\mathcal{P} }$ is Woodin in ${\mathcal{P} }$, and $o^{\mathcal{P} }(\kappa)=\delta$. ${\mathcal{P} }|\xi$ is the least active level of ${\mathcal{P} }$ above $\delta$.
• Figure 3.3.1: Hull of a stack of length $2$. $({\mathcal{M}}, {\mathcal{U}}, {\mathcal{M}}_1, {\mathcal{W} })$ is a hull of $({\mathcal{M}},{\mathcal{T}}, {\mathcal{M}}_2, {\mathcal{S}})$.
• Figure 3.3.2: Branch condensation for short tree strategies. Notations are as in Definition \ref{['branch condensation for short tree strategies']}. In the above, $\pi^{{\mathcal{T}},b} = \pi \circ \pi^{\mathcal{S}}_c \circ \pi^{{\mathcal{U}},b}$.
• Figure 3.7.1: $({\mathcal{T}},X)$ authenticates ${\mathcal{R}}$. The objects $\xi, {\mathcal{U}}$ etc. are as in \ref{['authentic lsp']}.
• Figure 12.7.1: Hypothesis of Lemma \ref{['lem:certified_extender']}

Theorems & Definitions (503)

• Definition 1.0.1
• Definition 1.0.2
• Remark 1.0.3
• Definition 1.0.4
• Definition 1.0.5
• Conjecture 1.0.7
• Definition 1.0.8
• Theorem 1.0.9: ATHM and Theorem \ref{['computation of hod']}
• Conjecture 1.0.10
• Definition 2.1.1