大基数和高阶度论

An important research program of recursion theory is to generalize the structural theory of Turing degree to ordinals (or cardinals). With the inspiration and help from Professor Woodin, in his recent research the applicant applied set theoretical ideas and techniques, extended the generalization to general definable degrees, which is expected to grow into a new research program ---- higher degree theory. The novelty of this new theory, besides exploring for and studying new types of degree structures, is to focus on the connection between the complexity of the degree structures of the countable cofinality singular cardinal and the strength of strong axioms (such as large cardinal axioms). The objective is to search for and characterize new degree structures, and to systematically analyze the correlations between the complexity of degree structures, the complexity of definable degree notions and the strength of strong axioms. The primary task of this project is to analyze the degree structures at countable cofinality singular cardinals in the least core model for infinite countably many Woodin cardinals.

递归论中一个重要的研究方向是将图灵度的结构理论推广到一般的序(基)数上。在Woodin教授的启发和帮助下，申请人在最新的研究中引入集合论的思想和方法，将这种推广推进到一般的可定义度上，发展出一个新的理论研究方向——高阶度论(Higher Degree Theory)。这套理论不仅仅是单纯地寻找和研究不同类型的广义的度结构，其新颖之处在于首次发现可数共尾度的奇异基数上的度结构的复杂程度与强公理(尤其是大基数公理)之间的内在联系。本次申请的课题是要沿着这个思路，寻找和刻画新型的度结构，系统地分析度结构的复杂程度、度定义本身的复杂程度以及强公理的强度之间内在的对应关系。本课题的首要任务是研究分析最小的含有无穷多个Woodin基数的内核模型中在可数共尾度的奇异基数上的度结构。

本课题研究高阶度结构与大基数之间的关系。项目执行期间，我们成功的获得了不可比Zermelo度现象在上至一个Woodin基数的经典内模型中的分布情况，获得极小度／极小覆盖现象出现的一些充分条件（猜测是必要的），还探索了关于反链和独立集的两个性质，这些是高阶度论研究的重要阶段性成果；此外在大基数方面还研究了\( I_{0}(\lambda) \)公理下\( L(V_{\lambda + 1}) \)里的组合性质，获得更多的与决策性公理下类似的组合性质。这里包括一篇发表在BSL的评论文章和两篇已投稿并收到推荐接收的审稿报告，基本完成了基金申请中研究计划。除此之外，我们还在非标准方法和Universally Baire集合的二分性质等方向做了一些探索工作并有所收获。