实数集合理想的基数不变量与力迫公理的关系

Cardinal invariants of sets of reals and forcing axioms are both important subjects in the research of axiomatic set theory. There were classical results which established the relationship between them. However，traditional forcing axioms usually made all these cardinal invariants large, namely the same as the continuum. As a result, the partial order among the cardinal invariants could not be utilized to analyze the type of reals in the model of forcing axioms. The distributive proper forcing axiom proposed recently behaves differently. The variety of partial orders among the cardinal invariants could be nicely preserved by the distributive proper forcing axiom. Moreover, this forcing axiom has certain strength of large cardinals and connects the low infinity of the reals and the high infinite structures..In this project, several new cardinal invariants of ideals on the reals will be considered. Based on existing work, the relationship between these cardinal invariants and the distributive proper forcing axiom will be investigated. The first step is to represent cardinal invariants suitably and find partial orders which are provable with the representation. The second step is to show the consistency of partial orders among these cardinal invariants, which will be realized by forcing various types of reals to the models. Then, the preservability of the distributive proper forcing axiom will be studied. New approaches will be considered so as to prove a hand of preservation theorems in the same way. Lastly, these results will be combined and applied on the relationship between cardinal invariants and the distributive proper forcing axiom..This project is supposed to develop the theory and techniques of forcing, to refine the hierarchy of forcing axioms and to achieve better understanding on the structure of the continuum. Research on large cardinals, inner model as well as analysis and set theoretic topology will benefit from this project.

实数集合上的基数不变量和力迫公理均是集合论的重要研究对象，有不少研究建立两者的联系。然而，传统的力迫公理将基数不变量极大归一化，限制了通过基数不变量的偏序关系来分析模型中实数类型。近年提出的分配性恰当力迫公理既较好地保持基数不变量的多样偏序关系，又保留一定大基数强度，从而能有效地建立实数的“小无穷”和大基数的“大无穷”之间的联系。.本项目基于已有工作，着眼于实数集合理想上若干新的基数不变量，深入研究其与分配性恰当力迫公理之间的逻辑关系。首先通过合理的表示方法实现基数不变量之间偏序关系的可证明性，然后通过实数力迫法构建模型实现偏序关系的协调性，接着由大基数力迫法探索新的路线实现分配性恰当力迫公理的可保持性，最后结合起来研究基数不变量与力迫公理的关系。.本项目的研究将丰富力迫法的理论与技术，细化力迫公理分层，深化实数结构解析。同时也有益于大基数、内模型等方向以及分析、集论拓扑等研究领域的发展。

基数不变量是特定数学性质的凝练，集合论从两方面研究与无穷集合相关的基数不变量。一方面是在公理系统下，研究如何对基数不变量描述、表示、计算和比较；另一方面是研究在不同的集合论模型下基数不变量的性质，特别是探讨如何通过力迫扩张构造新模型，保持部分基数不变量的同时改变其它基数不变量。从力迫法的应用中提炼出力迫公理。力迫公理形成良好的分层结构，可以作为通常公理系统的补充。本项目一方面在较弱的假设下，研究了不可数交换群上的理想所定义的若干个基数不变量，对其做了较完整的刻画分析和比较；另一方面研究了力迫公理直接应用的方法。已取得的成果开启了较新的研究角度，为后续研究做了准备。