基于幂容许覆盖的集合论端点扩张模型
End extending models of set theory via power admissible covers
摘要：
出于问题涉及端扩展模型的集合论，我们开发的基础设施的权力容许覆盖建设（在病态成立的模型的集合论），一个扩展的机器的容许覆盖发明的Barwise作为一个通用的工具，推广模型理论的结果可数良基模型的集合论可数病态成立的。我们的发展的权力容许机制，使我们能够获得新的结果，关于幂集保持结束扩展和秩扩展的可数模型的子系统的$\mathsf{ZFC}$。Kripke-Platek集合论$\mathsf{KP}$的规范扩展$\mathsf{KP}^\mathcal{P}$在我们的工作中起着关键作用;我们的一个结果通过证明$Σ_1^\mathcal{P}\text{-}\mathsf{Foundation}$在$\mathsf{KP}^\mathcal{P}$中是可证明的（不调用选择的选择公理）来改进Rathjen的一个定理。
英文摘要：
Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalizing model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of $\mathsf{ZFC}$. The canonical extension $\mathsf{KP}^\mathcal{P}$ of Kripke-Platek set theory $\mathsf{KP}$ plays a key role in our work; one of our results refines a theorem of Rathjen by showing that $Σ_1^\mathcal{P}\text{-}\mathsf{Foundation}$ is provable in $\mathsf{KP}^\mathcal{P}$ (without invoking the axiom of choice).
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