与大基数相关的力迫法及其应用

Forcing with large cardinal is one of fundamental approaches in foundation of mathematics for discovering and validating the consistency of new mathematical statements. The main objective of the project is to study the following two classes of forcing with large cardinals and their application. First. we will study forcing with models as side condition, and its applications in measuring property, P-ideal dichotomy, Martin's Axiom and special tree property. Second, we will define and study new Prikry-type forcing and its applications on the tree property.

与大基数相关的力迫法是数学基础研究中发现并验证新的数学命题相容性的核心方法。本项目主要研究如下两类特殊的与大基数相关的力迫法及其应用。首先，研究模型边条件力迫法，及其在measuring与连续统问题，P理想二分法，马丁公理与特殊树性质上的应用。其次，定义并研究新的Prikry型力迫，及其在树性质上的应用。

在项目中对与大基数相关的力迫方法进行了研究。与合作者一起证明了对任意自然数n，存在模型使得不可数线性序的基大小为n；解决了Larson-Todrcevic关于MA(S)[S]和K2关系的问题；证明了PFA与存在可定义非荟萃理想的相容性；证明了在无选择公理时，可能存在特殊集合上的平方到幂集的满射，说明Cantor的满射版本对角化定理是最优的。参与人游志兴与袁嘉辰合作解决了Bargaria-Magidor关于omega_1强紧基数的共尾数问题。