论文标题
设置理论的驯服$ II $
Tameness for set theory $II$
论文作者
论文摘要
该论文是两者中的第二本书，表明（假设大型红衣主教集理论是一种可拖延（并且我们敢于说驯服）一阶理论，当时是在一阶签名中形式化的，带有自然谓语符号，用于第二和三阶算术的基本可定义概念，并吸引了模型的模型完整性和模型完整性和模型伴侣的模型概念。 具体而言，我们使用将通用绝对性结果链接的一般框架来模拟第一篇论文中介绍的模型，以表明伍德丁的公理$（*）$的强大形式都意味着，任何理论$ \ t $延长了$ \ shatsf {zfc} $由合适的大型基准套件的合适的大型基准套件对某些型号$ t^*$ t^*$ t^* $ω$和$ω_1$的符号，这是$ω_1$上非平稳理想的谓词符号，某些Lightface的符号可定义的普遍baire集。 此外，$ t^*$由$π_2$ -Sentences $ψ$ for $τ$进行公理，因此$ t $证明了$$ l（\ mathsf {ub}）\ models（\ mathbb {p} _ \ max \ max \ max \ max \ max \ max \vdashψ表示包含普遍的baire集的最小的传递模型。 我们结果的关键是Asperò和Schindler的最新突破确定伍德丁的Axiom $（*）$的强烈形式来自$ \ Mathsf {mm}^{++} $。
The paper is the second of two and shows that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we use the general framework linking generic absoluteness results to model companionship introduced in the first paper to show that strong forms of Woodin's axiom $(*)$ entail that any theory $T$ extending $\mathsf{ZFC}$ by suitable large cardinal axioms has a model companion $T^*$ with respect to certain signatures $τ$ containing symbols for $Δ_0$-relations and functions, constant symbols for $ω$ and $ω_1$, a predicate symbol for the nonstationary ideal on $ω_1$, symbols for certain lightface definable universally Baire sets. Moreover $T^*$ is axiomatized by the $Π_2$-sentences $ψ$ for $τ$ such that $T$ proves that $$ L(\mathsf{UB})\models(\mathbb{P}_\max\Vdashψ^{H_{ω_2}}), $$ where $L(\mathsf{UB})$ denotes the smallest transitive model containing the universally Baire sets. Key to our results is the recent breakthrough of Asperò and Schindler establishing that a strong form of Woodin's axiom $(*)$ follows from $\mathsf{MM}^{++}$.
