通用支配与理想化力迫
Universal domination and idealized forcing
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中文总结 AI 辅助
该研究引入波兰空间可定义σ-理想的通用性质，将其作为理想化力迫固有性的基准并统一相关结果，借此回答了两个关于理想化力迫的问题。
中文摘要 AI 辅助
我们引入波兰空间上可定义σ-理想的通用性质，该性质一方面可作为由正Borel集按包含关系排序的相关理想化力迫的固有性的基准，另一方面统一了Zapletal著作中的诸多结果。我们证明，在温和的绝对性假设下，该性质蕴含固有性、各类二分定理，且在Solovay模型和AD⁺下对良序并封闭，此外还满足其他性质。该著作中研究的所有主要类别的固有理想化力迫均具有此性质。进一步，我们利用该视角回答了Khomskii的问题，表明用于添加最终不同实数或精炼实数的朴素理想化力迫在某些条件下并非固有；同时还回答了与Kanovei、Sabok和Zapletal提出的Borel集生成的σ-理想的可定义性相关的问题。
英文摘要
We introduce the universality property of a definable $σ$-ideal on a Polish space, which, on one hand, can serve as a benchmark for the properness of the associated idealized forcing of positive Borel sets ordered by inclusion, and, on the other hand, unifies many of the results that can be found in Zapletal's book. We show that under mild absoluteness assumptions, it implies properness, various dichotomy theorems, and closure under well-ordered unions in the Solovay model and under $\mathsf{AD}^+$, among other things. All major classes of proper idealized forcings studied in the book have this property. Further, we use this viewpoint to answer a question of Khomskii by showing that the naive idealized forcing for adding an eventually different real or a refining real is not proper below some condition. We also answer a question related to the definability of $σ$-ideals generated by Borel sets due to Kanovei, Sabok, and Zapletal.
